neutron data booklet
DESCRIPTION
Data booklet for neutron scatteringTRANSCRIPT
NEUTRON DATA BOOKLET
AFFILIATIONS
H. Börner, J. Brown, C. J. Carlile, R. Cubitt, R. Currat, A. J. Dianoux, B.Farago, A.W. Hewat, J. Kulda, E. Lelièvre-Berna, G.J. McIntyre, S.A. Mason,R. P. May, A. Oed, J. R. Stewart, F. Tasset and J. Tribolet- Institut Laue-Langevin, France
I. Anderson - Spallation Neutron Source, Oak Ridge, USA
D. Dubbers - University of Heidelberg, Germany
R. S. Eccleston, M. Johnson and C. C. Wilson - ISIS, RutherfordAppleton Laboratory, Unitied Kingdom
G. Lander - Institut für Transuranium Elements, Karlsruhe, Germany
H. Rauch - Atominstitut der Österreichischen Universitäten, Austria
R. B. Von Dreele - Los Alamos National Laboratory, USA
W. Waschkowski - Technische Universität München, Germany
Secretary: A. Mader
Institut Laue-Langevin BP 15638042 Grenoble Cedex 9 FRANCEE-mail: [email protected] Web: www.ill.fr
Second Edition, July 2003
Copyright © 2003 Institut Laue-LangevinPublished by licence under the OCP Science imprint, a member of the Old City Publishing Group
ISBN: 0-9704143-7-4
To order additional copies:Old City Publishing, 628 North Second Street, Philadelphia, PA 19123 USAPhone: + 1 215 925 4390 Fax: +1 215 925 4371 Web: oldcitypublishing.come-mail: [email protected]
NEUTRONSFOR SCIENCE
Preface for Neutron Data Booklet
1. Neutron Properties1.1. Neutron Scattering Lengths
A. Rauch and W. Waschkowski
1.2. The Neutron and its Fundamental InteractionsD. Dubbers
2. Neutron Scattering2.1. Small-Angle Scattering
R. P. May
2.2. ReflectometryR. Cubitt
2.3. Time-of-Flight Neutron DiffractionC. C. Wilson
2.4. Polarised Neutron ScatteringE. Lelièvre-Berna, J. R. Stewart and F. Tasset
2.5. Magnetic Form FactorsJ. Brown
2.6. Time-of-Flight Inelastic Neutron ScatteringR. S. Eccleston
2.7. Three-Axis SpectroscopyR. Currat and J. Kulda
2.8. The Basics of Neutron Spin-EchoB. Farago
2.9. Neutron Diffraction on ReactorsA. W. Hewat and G. J. McIntyre
CONTENTS
3. Techniques3.1. The Production of Neutrons
C. J. Carlile
3.2. Neutron OpticsI. Anderson
3.3. Detectors for Thermal NeutronsA. Oed
4. Activation and Shielding4.1. Activation Table of the Elements
M. Johnson, S. A. Mason and R. B. Von Dreele
4.2. Shielding of RadiationsH. Börner and J. Tribolet
5. Miscellaneous5.1. Radioactivity and Radiation Protection
5.2. Physical Constants
5.3. Properties of the Elements
5.4. Periodic Table of the Elements
NEUTRON DATA BOOKLET,SECOND EDITION
To make freely available a small pocketbook that covers the widefield of neutron scattering, is clearly an idea which was timely. The
first edition of 5000 copies was absorbed by the growing neutroncommunity worldwide within a few months of being published.
The obvious need for a second edition has allowed certain correctionsto be made to the first edition – inevitable given the speed with which itwas written and printed – and to rectify some omissions – notably anew chapter on diffraction methods on continuous sources by AlanHewat and Garry McIntyre.
I want to thank Christian Vettier, ILL’s Science Director, who is at theorigin of this project and the two editors Gerry Lander from Kalsruheand José Dianoux from Grenoble who have energetically pursued andimplemented the idea, and of course the authors who have once againresponded admirably to a tight deadline.
Colin CarlileILL Director16th March 2003
PREFACE FOR NEUTRON DATA BOOKLET
Welcome to the Neutron Data Booklet. The success of the X-rayand Nuclear Physics Booklets, and the ever-increasing number
of neutron users, has led the ILL in collaboration with Old CityPublishing, to compile this "little book of facts."
We are first grateful to Christian Vettier of the ILL who persuaded us toundertake this task and helped in many ways in getting people tocooperate. We thank all those who contributed most sincerely; werealise that this is not a research document, and therefore lacking in realexcitement. On the other hand, we hope they (and you the reader) willfind it above all "useful" and get to feel that having one in your pocketis part of the dress code for a practicing neutron scatterer. Our thanks tothe secretarial help at the ILL, who are listed. They gainfully struggledover tables to format and kept their cool.
Although this document was produced at the ILL, we recognise theimportance of spallation neutrons, and we hope you will find all youneed about neutrons in general in the pages. The editors would like tobe informed of both errors and omissions so that these may becorrected in future editions and for when the information is loaded ontothe web. In particular, suggestions for further tables or chapters will bewarmly received, especially if they are accompanied by a "volunteerauthor"!
Finally, thanks to Ian Mellanby and Guy Griffiths of Old CityPublishing for following through with this and making such apresentable final product.
Albert-José Dianoux, ILL, Grenoble, FranceGerry Lander, ITU, Karlsruhe, Germany
Neutron Scattering LengthsH. Rauch and W. Waschkowski
1. IntroductionFree neutrons are ideal tools for a lot of experiments, for instance for theinvestigation of the atomic and molecular structure and of the dynamics ofcondensed matter. Fission and spallation processes are used for an effectiveproduction of free neutrons. These neutrons can be extracted from the moderatorblock and are guided to the experimental devices outside the reactor shielding. Formost experiments monochromatic neutrons are needed, which have to be filteredout from the thermal spectrum leaving the moderator. When pulsed sources ormechanical choppers are used, the time-of-flight technique can be used to measurethe energy and the energy change of neutrons. Absorption, transmission andscattering of neutrons are also used for non-destructive inspection of materials.
The scattering length of the neutron-nucleus system is the basic quantitywhich describes the strength and character of the interaction of low-energyneutrons with the individual nuclei and atomic structures. The values of scatteringlengths vary irregularly from one nucleus to another due to their strong dependenceon the details of the individual nuclear interaction. Therefore, low-energy neutronsare an important tool for the investigation of the static and dynamic properties ofcondensed matter since they distinguish between various elements and isotopes.They can also be used for a detailed study of the interaction of the neutron as anelementary particle with its surroundings. The related scattering lengths are offundamental interest for structure and dynamics investigations of condensedmatter, in nuclear research, for biological systems, and for other disciplines. Inmost cases accurate and reliable values of scattering lengths for chemical elementsand also for separated isotopes are needed as input data for the interpretation ofexperiments with neutrons.
In the past various tables of scattering lengths have been presented in theliterature, sometimes with the aim to collect all experimental results including thereferences [1-6]. Thermal cross section values have been added because they are
1.1-1
NEUTRON PROPERTIES
also relevant in scattering experiments when extinction and background effectshave to be estimated [2,3,5,7].
2. Properties of the neutronNeutrons have distinct particle properties, which influence the experimental scat-tering results. They have nearly no electrical properties: “no” electrical charge, “no”electrical dipole momentum. Neutrons mainly obey nuclear interaction. However,their magnetic moment couples to the local magnetic field of magnetic atoms andions. Neutrons also exhibit weak interaction which is responsible for neutron decay.
Table 1: Neutron PropertiesMass m = 1.674928(1) · 10-27 kgSpin s = -h/2magnetic moment µµ = -9.6491783(18)⋅⋅10-27 JT-1
ββ-decay lifetime ττ = 885.9 ±± 0.9 sconfinement radius R = 0.7 fmquark structure udd
3. Scattering lengthsLow-energy neutrons are scattered isotropically within the center-of-mass system,which indicates that no orbital momentum is involved in the scattering process (l =0). This fact is equivalent to the statement that the range of interaction is muchsmaller than the center-of-mass wavelength, λ, of the neutrons. This enables, withinthe Born approximation and the ordinary scattering theory, the introduction of apoint-like interaction in the form of the Fermi pseudopotential [8]
(1),
where mr = mnmk / (mn + mk) is the reduced mass of the neutron (mn) – nucleus system.Here, a is the free scattering length, which is related to the scattering amplitude fo of thescattered spherical wave by fo = -a. This definition is chosen to get for an infinite repul-sive potential the relation a = R, where R is the radius of this scattering potential.
Rather well known is the scattering of a plane neutron wave (wave number k)by a single fixed nucleus (spin I = 0) which results in a spherical scattered wave:ƒ(θ) exp(ikr)/r. Within the Born approximation the scattering amplitude does in thiscase not depend on the scattering angle and one gets
. (2)More rigorously, for slow neutrons, s-wave scattering dominates and leads to an s-wave phase shift δo. The scattering amplitude fo, is a complex quantity given by
f a( )θ = −
Vmr
( )r h rr r
2 ( )= π δ
2
a
1.1-2
(3)
This relation shows that the scattering length determines the leading term of how thephase shift depends on the momentum of the interacting particle. Thus, in the lowenergy limit, the relation holds a ≅ δo/k.
The total scattering cross section σs is given by the imaginary part of thescattering amplitude by the optical theorem.
(4)
The scattering amplitude consists of a potential and a resonance part, which can bedetermined from resonance parameters.
(5)
By Breit-Wigner formalism the sums can be written as
(6)
where the summation must be carried out over all resonances r (all means resolved andunresolved resonances as well as bound levels at negative energies) at the resonanceenergies Er. Γnr stands for the neutron scattering width and Γr for the total (scatteringand absorption) width.
For nuclei with spin I 0, the s-wave neutron interaction leads to twocompound spin states (with I + 1/2 and I – 1/2 ). Then the resonance sums must beweighted with the spin statistical factors g±.
(7)
The potential part of the scattering amplitude is the spin independent potentialscattering radius R' being of theoretical interest (Fig. 1). Systematic calculations ofthe neutron-nucleus scattering parameters from experimental data have beenreported by Aleksejew et al. [9].
Neutron optical phenomena occur due to the collective interaction with manyscattering centers. Therefore, the mean phase shift <δo>, or the mean interactionpotential , become relevant. In this case the momentum transfer occurs to the wholeassembly of particles and therefore, the center of mass system equals the laboratory
= I +1
2I +1; g =
I2I +1
= g + g .
-
+ -
g+
+ −
;
, , ,Σ Σ Σ1 2 1 2 1 2
ΣΓ
Σ Γ ΓΣ ΣΓ Γ1 2 = , = r r
nr r
r r r
nr r
r
kk
E E
E E
kk E Er r
−( )−( ) + −( ) +2
4
2
4
2 2
2/
Re(
sin cos
Im( cos sin
f
f
o
o o
o o o
) = f + f
1
2k 1-
) = 1
2k 1-
p r
o2
o2
= ( ) −
− ( ) −
Σ Σ
Σ Σ
2 2
1 2 2
1
1
δ δ
δ δ
σ πs of =
4k
Im( )
f f ao o =
1
2ik (e -1),
o
-2i oδRe( )=−
1.1-3
system and the bound scattering length b, where b = a (mn + mk)/mK. Its averagevalue denotes the coherent scattering length bc.
(8)and its variance determines the incoherent scattering length bi
(9)The related cross sections are
(10)and the total scattering cross becomes
(11)With the kinetic neutron energy E = h2/2mnλ2 the refraction index n - being theratio of the wave number inside (K) and outside (k) of a mean potential of anuclei collective with a density N - follows from Eq. (1) and can be expressed as:
, (12)
which lead to a refraction index of
(13)In this simple treatment absorption and magnetic effects are neglected.
When the absorption (reaction) processes are included one obtains a complexindex of refraction in the form [10,11]
(14)where σr = σa + σi includes the absorption and incoherent scattering cross section.
nN N
1 - b - 2
i2
c2 r r=
+λλ
σλ
σ λπ2 4
2
nV
E
Nbc=K
k= 1 1− ≅ − λ
π2
2
n Nbc
2 21− − −=VE
= D /π
V
σ σ σt c i= +
b g bi2 2 g - b-= ( )+ − +
b g b g bc = ++ + − −
1.1-4
Figure 1: Measured and calculated values of the potential scattering radius [2,3].
σ π σ πc c2
i2 4 b 4 b= =; i
4. Methods of measurementsScattering lengths can be deduced from total cross section measurements, but forthis evaluation a lot of corrections are necessary. Therefore, techniques for a moredirect determination of coherent scattering lengths have been developed which arebased on neutron optical phenomena. A great variety of methods for thedetermination of coherent scattering lengths exists, because all collectiveinteraction effects are characterized by this quantity. The standard method is basedon Bragg diffraction; whereas advanced methods use classical neutron optics,where a more direct relation between the scattering length and the measurablequantities exist [4,11]. That means, less corrections must be performed during thedata evaluation and the accuracy of the deduced quantity becomes much higher.The considerable efforts to determine scattering lengths with high accuracy forinvestigations in fundamental questions lead to techniques of single crystal neutroninterferometer [12] and of gravity refractometry [13]. With both techniques areliability of the order of ∆b/b = 10-4 can be achieved.
Advanced methods for measurements of coherent scattering lengths are totalreflection, Christiansen filter technique, and neutron interferometry. The spindependent scattering lengths, which are of particular interest for theoreticalinterpretations and for isotopic substitution techniques, can be obtained by polarizationtechniques of neutrons and nuclei. Incoherent nuclear polarization-dependent crosssections can be determined from a combination of free scattering cross sections withdirectly measured scattering lengths as well. Free potential scattering radii, which areof fundamental interest for the theory of optical nuclear model, can be deduced fromcoherent scattering lengths substracting the resonance contribution. For moreinformation about these techniques we refer to literature [6,11].
5. Data of coherent scattering lengths and of thermal cross sectionsThis new compilation has the aim to give recommended coherent scattering lengthsfor elements and if available for isotopes; this will be done including recommendederror bars according to the best possible criteria such as check of consistency of thecomplete nuclear data set (absorption, energy dependence of scattering crosssection, incoherence, spin depending scattering lengths). Table 2 shows therecommended values. For the spin dependent scattering length an alternative set ofb+ and b- values exists because in most cases (b+ - b-)
2 has been measured. Column7 of Table 2 indicates whether separate (b+ - b-) values are known (+/-) or/andwhether a strong energy dependence of the scattering lengths exists (E). The tablegives the most probable set. The explanation of the symbols is as follows:
1.1-5
Column1: ZSymbA nuclide: charge number Z - element symbol - mass number A2: p or T1/2 abundance (in %) or half-live T1/2
3: I nuclear spin I4: bc bound coherent scattering length (in fm)5: b+ spin-dependent scattering length for I+1/2 (in fm)6: b- spin-dependent scattering length for I-1/2 (in fm)7: c indication whether separate (b+ - b-) values are available (+/-)
and/or a strong energy dependence exists (E)8: σcoh coherent cross section in barn9: σinc incoherent cross section in barn10: σscatt total cross section in barn11: σabs thermal absorption cross section in barn for 0.0253 eV* stands for estimated values
All scattering cross sections follow from known scattering lengths (Eqs. (8)-(11)) but in the table separately measured values are given which results in a certaininconsistency of the values reflecting the fact of conflicting experimental results.
A summary of all measured neutron scattering lengths until 1991 is given in[1] and the references for the values recommended in this summary can be found in[6]. It should be mentioned that there is still a lack of information of scatteringlengths of many isotopes and especially of many spin dependent scattering lengths.Any reader is asked to inform the authors about new or unnoticed values. Thankyou in advance.
The authors acknowledge the valuable support of M. Hainbuchner andE. Jericha in course of data handling.
References[1] Koester, L., Rauch, H., Seymann, E.: Atomic Data and Nucl. Data Tab. 49
(1991) 65.[2] Mughabghab, S.F., Divadeenam, M., Holden, N.E.: Neutron Cross Sections,
Vol. 1, Part A; Academic Press, New York, London (1981).[3] Mughabghab, S.F.: Neutron Cross Sections, Vol. 1, Part B; Academic Press,
New York, London (1984).[4] Sears, V.F.: Meth. Exp. Phys. 23A (1986) 521.[5] Sears, V.F.: Neutron News 3 (1992) 26.[6] Rauch, H., Waschkowski, W.: Landolt-Boernstein 16A, Chapt. 6, Springer
Berlin (2000).
1.1-6
[7] CINDA 2000, "Microscopic Neutron Data". IAEA, Vienna 2000, ISSN1011-2545.
[8] Fermi, E., Marshall, L.: Phys. Rev. 71 (1947) 666.[9] Aleksejev, A., Barkanova, S., Tambergs, J., Krasta, T., Waschkowski, W.,
Knopf, K.: Z. Naturforschung 53a (1998) 855.[10] Goldberger, M.L., Seitz, F.: Phys. Rev. 71 (1947) 294.[11] Sears, V.F.: Neutron Optics, Oxford University Press (1989).[12] Rauch, H., Werner, S.A.: Neutron Interferometry, Clarendon Press, Oxford
(2000).[13] Koester, L., in: Neutron Physics (Ed. G. Hoehler); Springer Tracts in Mod.
Phys. 80, Springer Berlin (1977).
1.1-7
1.1-8
1.1-9
1.1-10
1.1-11
1.1-12
1.1-13
1.1-14
1.1-15
1.1-16
Small-Angle Scattering
R. P. May
IntroductionSmall-angle scattering (SAS) deals with the deviation of electromagnetic or particlewaves by heterogeneities in matter. Here, we shall treat the case of neutrons.
SAS allows one to study objects and structures in a size range of about 10 to1000 Å at low resolution only, but on the other hand, it is not necessary to dispose ofcrystals. The samples can be considered as composed of elementary volumes with alinear dimension of a few Å, i.e. about the dimension of the neutron wavelengthsused. The scattering then depends on the scattering length densities of the elementaryvolumes, which can readily be influenced for neutrons by modifying the isotopiccomposition of the sample. This technique is known as “contrast variation” and haslargely determined the success of small-angle neutron scattering (SANS) in the fieldsof soft condensed matter and biological structures. Other important reasons for usingneutrons rather than X-rays are the natural differences in scattering length density forcertain classes of matter, the ease with which one can use bulky sample environmentand thick samples, and of course, the sensitivity of neutrons for nuclear spins.
Small-angle scattering principlesWhen a plane wave in z direction (Ψ = eikz) hits a sample, spherically symmetricalwaves, Ψ = - (b/L) eikL are emitted from the elementary scatterers in the sample, i.e. thenuclei; L is the distance of an observation point from the sample. The complex scatteringamplitude b has the dimension of a length; it is called scattering length. The sphericalwaves interfere and create a pattern on the neutron detector. The scattering geometry issketched in Fig. 1.
2.1-1
NEUTRON SCATTERING
When the sample is disordered, as in the majority of the applications of SANS,the scattering pattern is isotropic, and the scattering can be expressed as a functionof the modulus of Q: Q = |Q| = (4π/λ) sin θ ≈ 2πr/(λL). λ is the wavelength of theneutrons, L the sample-to-detector distance, and r the distance of the spot of thescattered beam on the detector from that of the direct beam. The sample can be ori-ented by its preparation procedure or by external fields, in particular by a magneticfield, e.g. for studying flux lines, or by a shear gradient.
Scattering cross sections and absolute intensityThe number of neutrons scattered by a sample into a solid-angle element ∆Ω of adetector during a time ∆t at a momentum transfer Q can be expressed as
(1).
Φo is the neutron flux (neutrons per cm2 per second) on the sample. Φo·∆t is pro-portional to the counts in the monitor counter used for determining the duration of
I Q tTT
a Dd
dQs o
s
aps
V( ) = ( ) ∑Φ ∆Ω ∆Ω
ε λ ( )
2.1-2
Figure 1. Scattering geometry:An incoming plane wave (wave vector k0) hits two elementary volumes, 1 and2. They emit spherical waves (scattering vector k1) that, at an angle 2θθ, are de-phased due to the different distances of the two volumes from the wave front.The momentum transfer Q is equal to Q = k1 - k0.
a scattering run. Ts/Tap is the sample transmission, i.e. the ratio of the direct beamintensities behind and in front of the sample, respectively. a and Ds are the illumi-nated area and the thickness of the sample, respectively, ε(λ) the detector efficien-cy for a given wavelength, and dΣV/dΩ is the differential scattering cross sectionper unit volume.
dΣV/dΩ can be expressed as
(1a)
In the case of isotropic scattering, one averages all the possible configura-tions of the nuclei in the sample, expressed by <…>:
(1b)
Using an attenuated beam intensity with known attenuation factor fa, weobtain from equation 1
(2)
for data that were divided by a background-corrected water run (see below) with
(2a)
A broad treatment of absolute scaling techniques in SANS can be found inWignall and Bates (1987).
The differential scattering cross section per unit mass of the sample,dΣV(Q)/dΩ, is
(3),
with cs, the sample concentration, in g·cm-3. Finally, the differential scatteringcross section per particle, dΣM(Q)/dΩ, is
(4),
when the molecular mass M is known; NA is Avogadro's number.
Molecular massThe molecular mass M can be obtained by considering
dd
Qd Q
dMN
K Mc D T N
I QI Q
M m
a s s s A
s
w
ΣΩ Ω
( ) = ( ) = ( )( )
∑
dd
Qc
d Qd
Kc D T
I QI Q
m
s
V
s s s
s
w
ΣΩ Ω
( ) = ( ) = ( )( )
∑1
K IT
f a tw
ap
a o
= ( )( )
00ε λ, 'Φ ∆Ω ∆
dd
QK
D TI QI Q
V
s s
s
w
ΣΩ
( ) = ( )( )
Φo'
d
dQ
Vb eV
j
i i j∑ ( ) = ∑< −( )>
Ω b i
i, j
1 Q r r
d
dQ
Vb b
je
V
i
i i jΣΩ
i, j
( ) = ∑−( )1 Q r r
2.1-3
(5)
as(6)
,with , the (dry) partial specific volume of the particle.
Background correctionsThe background in a sample is composed of a transmission-dependent part (sampleholder, e.g. a quartz cell, and matrix scattering, e.g. incoherent scattering from aproton-containing buffer solution) and a transmission independent, but also time-dependent part (electronic and neutron noise).
Of the former, the term that is proportional to the transmission (sample con-tainer scattering) is easy to treat, whereas the incoherent term (proportional to 1-T)is more complicated. Since the empty-cell contribution is weakened by the trans-mission of a water sample, the empty-cell and noise contributions are corrected by
(7)
Mw, Me and MCd are the neutron monitor counts of the water sample, the containerand the noise measurements (for identical geometry, wavelength etc.), and Mo is areference monitor for scaling. Tw is the water transmission, i.e. the ratio of theintensity transmitted by the water sample and that of the empty cell.
In the more general case of subtracting a purely coherent container or referencebackground from a sample, we get
(8)
Ts and Tb are defined according to Tw above.
Debye formulaIf the scattering is produced by a dilute ensemble of particles of limited size, thescattering curve can be approximated by the first terms of a Taylor series. According to Debye (1915), <eiQr> = (sin Qr)/Qr for randomly oriented particles.The scattering from N particles of volume VP can be written as a sum of contribu-tions from elementary volumes i, j with scattering densities ρi and ρj, separated bya distance rij:
I Q MI QM
TT
I QM
TT
I QMs
s
s
s
b
b
b
s
b
Cd
Cd( ) = ( ) − ( ) − −
( )
0 1
I Q MI QM
TI QM
TI QMw o
w
ww
e
ew
Cd
Cd( ) = ( ) − ( ) − −( ) ( )
1
vp
Md Q
dN
v
m a
p p
= ( )( )
∑
Ω ∆ρ2
dd
Q b V v m vM
NM
ii
p p p p p p pA
ΣΩ
∆ ∆=( ) =
= ( ) =
−∑0
22
2
ρ ρ ρ
2.1-4
(9)
Radius of gyrationSince (sin x)/x ≈ 1 - x2/6 + x4/120 - x6/5040 +... , one can express I(Q) near Q = 0 as
(10),
where in analogy to mechanics RG is the radius of gyration of the particle or theweight average of the radii of gyration of a mixture of particles.Eq. 10 is approximated again in two ways, the Zimm (1948) approximation for1/(cs I(Q))
(10a)
and the Guinier (1955) approximation that uses the similarity of the first terms ineq. 10 with the expression e-x ≈ 1 - x + x2/2 - ... to write
(10b).
Expression 10b was historically very useful, because it allowed one to extract RGby plotting I(Q) vs. Q2 on logarithmic paper.
RG can be obtained without the need of absolute calibration. In practice, equa-tions 10a and 10b do not yield the same value of RG, because the equations are notidentical, and because the fit cannot start at Q = 0 due to the beam stop.
Table 1. Radii of gyration (RG) of simple triaxial bodies (Mittelbach, 1964)body RG
sphere (radius R) (3/5) R2
hollow sphere (radii R1 and R2) (3/5) (R25 - R1
5) / (R23 - R1
3)ellipsoid (semi-axes a, b, c) (a2 + b2 + c2) / 5parallelepiped (edge lengths A, B, C) (A2 + B2 + C2) / 12elliptic cylinder (semi-axes a, b; height h) (a2 + b2) / 4 + h2/ 12 = Rc2 + h2/ 12hollow cylinder (radii R1 and R2; height h) (R1
2 + R22) / 2 + h2/ 12
Rc is the cross-sectional radius of gyration
ModellingFrequently, scattering data are analysed by comparing the experimental data
with that from models using least-squares fitting procedures, including polydisper-
I Q I Q e
QRG
( ) ≈ =( )−( )
0
2
3
cI Q
cI Q
QRs s G
( )≈
=( )+ ( )
01
3
2
...
I Q I QQR QRG G( ) = =( ) − ( ) + ( ) −
0 1
3 60
2 4
...
d
dQ
N
Vd d
V
i jij
ij
i jvp
ΣΩ
( ) = ( ) ( )∫∫ ρ ρr rQr
Qrr r
sin
2.1-5
sity treatment and instrumental smearing. The method and a catalogue of analyti-cal and semi-analytical expressions is described in Pedersen (1997).
Some form factors for simple bodies are listed in table 2.
Table 2. Form factors for simple bodies
Sphere of radius R
Two-shell sphere
where fi and Ai are the relative scattering weights and amplitudes of the outerand inner shells,
cylinder/disk
where 2d is the thickness, R the disk radius, J1 the first order Bessel func-tion, and 0- is the angle between the disk normal and Q)
Membrane, thickness 2d
Pair-distance distribution functionEquation (9) can be rewritten to yield
(11).
γ(r) is the correlation function (Debye and Bueche, 1949). Introducing p(r), the pair-distance distribution function, p(r) = r2Vγ(r), one gets
(11a)
The Fourier inversion of 11a,
(12)
can, in general, not be solved, because the scattering function is neither known atQ → 0, due to the beam stop, nor at Q → ∞, due to the limited maximum scatter-
p rd
dQ Qr Qr drV
o( ) sin= ∑∫ ( ) ( )∞1
2 2π Ω
d
dQ p r
Qr
QrdrV
o∑ ( ) = ( )∫ ( )∞
Ω4π sin
.
d
dQ r V r
Qr
QrdrV∑ ( ) = ∫ ( )∞
Ω 4 2
0π γ( )sin
P QQ
Qd emQd( ) = − ( )
−( )21 22
22
cos /
P QQ d
Q d
J Q R
Q Rdd ( ) = ( ) ( )
∫ sinsin cos
cos
sin
sin/ θ
θθ
θθ
θπ 2 10
22
f V V V f f1 1 1 1 2 1 2 2 11= + −( )[ ] = −ρ ρ ρ ρ/ ,
P Q f A Q R f A Q R( ) = ( ) + ( )( )1 1 1 2 2 22
, ,
P Q A Q RQR QR QR
QRS( ) = ( ) = ( ) − ( )
( )
2
3
2
3,sin cos
2.1-6
ing angle and bad statistics at high Q. Glatter (1981) has introduced an “indirectFourier transformation”, a least-square method that overcomes this problem for par-ticles of limited size rmax.The radius of gyration can be calculated as the second moment of p(r),
(13)
Size distributionsIn the case of a dilute solution of monodisperse homogeneous particles with formfactor P(Q),
. (14)
n is the number concentration of the particles (number per unit volume), bi are thescattering lengths of the nuclei of the particle, ρb is the scattering-length density ofthe bulk solvent, and V the volume which circumscribes the particle itself and itsassociated perturbed solvent (or matrix).
For a size distribution of homogeneous particles that differ only in their dimen-sion, not in their shape, we can write
(15),
where Dn(R) is the number distribution of particles, i. e. the number of particles ofsize R per unit volume.
One can also define a volume distribution Dv(R) ∝ R3Dn(R), i.e. the sum ofvolumes occupied by particles of dimension R per sample unit volume, and anintensity distribution Di(R), i.e. the sum of intensity contributions per particle ofdimension R per sample unit volume.
Porod’s lawsPorod (1951) has shown that the total small-angle scattering from a sample, irrespec-tive of the way its density is distributed, is a constant, called Porod's invariant C:
. (16)
In practice it is difficult to measure the data in all the Q range necessary to evalu-ate Porod's invariant, but in certain cases it is possible to approximate the low-Q(Guinier) and high-Q limits.
High-Q limitAlso to Porod, we owe an asymptotic law which is valid for high values of Q (Q >>
C I Q Q dQ Vo= ( )∫ =∞ 2 2 22π ρ∆
d
dQ D R V R P Q R dRs
b no∑ ( ) = ∫ ( ) ( ) ( )∞
Ω∆ρ2 2 ,
dd
Q n b V P Q n V P Qsi b b
∑∑( ) = −( ) ( ) = ( )
Ω∆ρ ρ2 2 2
Rp r r dr
p r drG
r
r2
20
02= ∫
∫( )
( )
max
max
2.1-7
1/D), if there are sharp boundaries between the phases of the system. This is a con-dition that barely holds. The specific surface Os = S/V, with S, the particle (orstructure) surface, and V, the particle (or structure) volume, is given by
(17).
Due to the usually very high incoherent scattering of neutrons at high Q, it is oftenvery difficult, if not impossible, to determine Os with sufficient precision. On theother hand, plotting IQ4 versus Q4 ("Porod plot"), which allows one to calculateOs from the Q4 = 0 intercept, can help us to determine a residual background,which shows up as a constant slope for large Q.
ReferencesDebye, P. (1915) Ann. Phys. 28, 809-823.
Guinier, A., and Fournet, G. (1955) “Small Angle Scattering of X-rays”, New York:Wiley Interscience.
Mittelbach, P. (1964) Acta Phys. Austriaca 19, 53-102.
Pedersen, J.S. (1997) Adv. Coll. Interface Sci. 70, 171-210.
Wignall, G., and Bates, J. (1987) J. Appl. Crystallogr. 20, 28-40.
Zimm, B.H. (1948) J. Chem. Phys. 16, 1093.
General reading“Small-Angle X-ray Scattering”, Glatter, O., and Kratky, O., eds., London: AcademicPress (1982).
“Neutron, X-ray and Light Scattering: Introduction to an Investigative Tool for Colloi-dal and Polymeric Systems”, Lindner, P., and Zemb, Th., eds., Amsterdam: NorthHolland (1991).
Williams, C., May, R.P., and Guinier, A. (1994) "Small-Angle Scattering of X-rays andNeutrons" in “Characterisation of Materials”, E. Lifshin ed., Vol. 2B of "MaterialsScience and Technology", Weinheim: VCH Verlagsgesellschaft, pp. 611-656.
OSV
I Q Q
CsQ= =
( )[ ] →∞π
4
2.1-8
Reflectometry
R. Cubitt
IntroductionNeutron reflectometry is a technique for the study of planar structures with a widevariety of materials from magnetic multi-layers to biological systems at a solid-liquid interface. The method not only permits the derivation of material structuresperpendicular to the plane but also to ascertain the perfection of these layers at aboundary which maybe modified by roughness or inter-diffusion. Many problemssuch as the behaviour of lipid bi-layers in biological cells or the influence ofmagnetic impurities in superconductors can be greatly simplified by synthesisinglayers of the material on a substrate thus reducing the dimensionality of the system.
Reflectivity from a Thick SubstrateA beam of neutrons with its associated wave properties can behave in exactly thesame way as light when impinging on a mirror like surface. Provided any deviationof the beam is far from satisfying the Bragg condition of the crystaline structure,the neutron can be considered to interact with a constant potential Vo simply relatedto the coherent scattering length by the relation
(1)
where mn is the mass of the neutron, ρ is the number density of atoms in thematerial and b is the average coherent scattering length for these atoms. Theproduct bρ is known as the scattering length density often denoted by Nb. Mostmaterials have a positive b so in a positive potential a neutron has less kineticenergy and hence a longer wavelength (opposite to light where the wavelengthshortens). We may now consider what happens to a beam approaching a surfacewith a bulk potential Vo, infinitely deep.
With no structure within the surface the only potential gradient and henceforce is perpendicular to the surface. Only the normal component of the incomingwave vector, ki is altered by the barrier potential and it is the normal component ofthe kinetic energy Ei⊥ which determines whether the neutron is totally reflectedfrom the barrier or not.
V
mbo
n
=2 2π
ρh
2.2-1
(2)
If Ei⊥ <Vo then there is total reflection and the critical value of wave vectortransfer, qc will be when Ei⊥ =Vo giving:
(3)Assuming the interaction is elastic then conservation of momentum means that i.e.
q N q kc b i i= =16 2π θ as sin
Ek
mii i
n⊥ =
( )
h sinθ2
2
2.2-2
Figure 1: (a) Reflection of an incident beam from an ideally flat interface. kiand kr are the incident and scattered wave vectors, with angles θθi = θθ0 = θθ inthe incidence plane; q is the wave vector transfer; Nb1 is the scattering lengthdensity of the semi-infinite substrate. On the right is the scattering lengthdensity profile as a function of depth; (b) Simulation of the specular neutronreflectivity as a function of q from a silicon/air interface (Nb1=2.07x10-6Å-2).
the reflection is specular provided the sample is static. Any off specular reflectionmust be a result of potential gradients within the xy plane of the surface.
If Ei⊥ >Vo then the reflection is not total the neutron can either be reflected ortransmitted into the bulk of the material. The transmitted beam, kt with its normalcomponent of kinetic energy reduced by the potential must change direction i.e. itis refracted. The change in the normal wave vector is:
(4)
allowing us to define a relative refractive index n:
(5)
where λ is the neutron wavelength. For a material such as Si, Nb is 2.07x10-6Å-2 ,much less than one, so for thermal neutron wavelengths we can make a goodapproximation for n with the well known result:
(6)
This confirms the earlier statement about the wavelength change in the bulkbeing opposite to that of light (for positive b) as we see that n is less than one. Thetransmitted beam refracts towards the mirror plane and exactly at the point of totalreflection, the refracted beam travels along the surface.
All of the above discussion (apart from treating neutrons as waves) can bederived from classical physics. In order to describe all the physical aspects ofreflectometry we must use a quantum mechanical approach. The wavefunctiondescribing the probability amplitude of a neutron near to the surface is
(7)
Solutions for this above and below the surface are:
(8)
where r and t are the probability amplitudes for reflection and transmission.Continuity of the wavefunction and its derivative give the expressions:
(9)1 1+ = −( ) =⊥ ⊥r t k r t ki t
Ψzik z ik z
zik ze re Y tei i t= + =⊥ ⊥ ⊥− and
∂δ
22 2
220
2Ψzz
niz
k km
E V k+ = = −( ) −⊥ ⊥ where h
//
nNb≈ −1
2
2λπ
nk
k
k k N
k
N
k
Nt
i
i i b
i
b
i
b22
2
2 2
2 2
241
41= =
+ −= − = −⊥// ( )π π λ
π
k k Nt i b⊥ ⊥= −2 2 4π
2.2-3
which leads directly to the classical Fresnel coefficients found in optics:
(10)
In reflectometry we measure the reflectivity as a function of wavevector transfer orq. Using the expressions (3), (4) and (10) we can relate the measured reflectivity Rto q and qc. Note that what we measure is an intensity and thus is a function of thequantum mechanical probability squared.
(11)
When q>>qc this reduces to:
(12)
which is the reflectivity used in the Born approximation [1]Returning to the wavefunction within the surface (8) and using (4) we find that
when EI<V (or ki⊥ <4πNb or q<qc) we have a real solution of the form:
(13)
This is a very important result as it shows that even when the potential barrieris higher than the particle energy normal to the surface it can still penetrate to acharacteristic depth of . This 'evanescent' wave travels along the surfacewith wave vector k// and after a very short time is ejected out of the bulk in thespecular direction. Taking the value of Nb for Si again this penetration is of theorder of 100Å at q=0 rising rapidly to infinity at q=qc. No conservation laws arebroken, as the reflectivity is still unity due to the fact that this wave represents noflux transmitted into the bulk. The result also explains why when a thin layer ofmaterial (<100Å) such as Ni, which has an Nb twice as large as Si, is put onto a Sisubstrate we find that qc is still defined by the Nb of Si and not of Ni. The reasonbeing in this case the layer is thinner than the characteristic penetration depth andthe neutrons tunnel through the barrier.
Calculating the reflectivity from such a system or one with many layersrequires a general technique such as the optical matrix method [2,3,4,5]. Thetransmission and reflection from one layer to the next can be described as a matrixmultiplication product for each layer. The problem of inverting a reflectivity curveto extract Nb as a function of depth is complex and many profiles can produce the
q qc2 2
1 2−( )− /
Y te tez
i k N z q q zi b c
= =⊥−( ) − −( )2 1 2 2 2 1 2
41
2π/ /
Rq
Nb≈16 2
42π
R rq q q
q q q
c
c
= =− −( )+ −( )( )
22 2 1 2
2 2 1 2
2/
/
rk k
k kt
k
k ki t
i t
i
i t
=−+ +
⊥ ⊥
⊥ ⊥
⊥
⊥ ⊥
and =2
2.2-4
same reflectivity curve. A useful review of the various techniques for inversion canbe found in [5].
AbsorptionIn reality many materials have a finite absorption cross-section. This is dealt withby adding an imaginary part to the coherent scattering length.
btotal=bcoherent+ibabsorption (14)
It can be seen then from (8) that the transmitted and the reflected intensity isexponentially reduced by the presence of absorption. Even in the regime of total
2.2-5
Figure 2 (a):As Figure 1a with an additional thin layer of scattering lengthdensity Nb2. (b) Simulation of the specular neutron reflectivity as a functionof q from a 400 Å layer of nickel on a silicon substrate (Nb1=2.07x10-6Å-2,Nb2=9.04x10-6Å-2).
reflection, as the evanescent wave exists in a surface region of the material,absorption occurs reducing the total reflection amplitude to less than one. It shouldnot be forgotten that even materials such as Cd and Gd that are very strongabsorbers of neutrons can still have significant reflectivities.
Magnetic MaterialsWe have seen that a neutron in the bulk is affected by a mean potential related to thecoherent scattering length. If the material is magnetised then there is an additionalpotential associated with the interaction of the magnetic dipole moment of the neutronand variations of magnetic flux density B. This potential has the value:
(15)
where µ is the magnetic dipole moment of the neutron and B(r) is the spatiallyvarying magnetic field. If the incoming beam is polarized up or down with respectto the magnetisation of the sample then the magnetic potential switches sign. Themagnetic potential can be expressed in the form of a coherent scattering length:
(16)
where S is the effective spin of the magnetic atom perpendicular to the momentumtransfer of the reflection. The total coherent scattering length becomes polarizationdependent with the plus sign corresponding to a beam polarised by a polarizer witha magnetisation in the same direction as the magnetisation of the sample:
(17)
Assuming the sample is a saturated ferro-magnet, to extract the nuclear andmagnetic parts of b two measurements must be made, one with the polarizationparallel to the magnetisation, R+ and the other with the polarization inverted, R- byuse of a flipper [6]. A very important point in polarised neutron reflectometry(PNR) is to remember that magnetic reflection is only going to occur at a potentialboundary and from (15), that requires a step in B. If a sample has been magnetisedwithin the plane of a magnetic layer then Maxwells equations tell us that the step inB is µoM at the surface, where M is the magnetisation density of the layer. If thesample is magnetised normal to the plane then B is continuous at the boundary andthere is no potential step. Magnetic reflection can only occur in the presence of in-plane magnetisation components. Components of magnetisation normal to theplane do not reflect an in-plane polarized beam but can cause spin flip i.e. acompletely polarized beam will have a certain fraction polarized in the opposite
b b btotal nuclear m= ±
be
mSm
e
= 1 9132
.
V B rmag = − •µ ( )
2.2-6
direction after reflection. This can only be measured by the use of a second flipperand polarizer or analyser. Figure 3 shows the four combinations of the initial andfinal polarization states needed to measure the four reflectivities i.e. R++, R– –,R+ –, R– +. It should be noted that from the polarizer to the analyser there must be asmall vertical guide field that maintains the polarization on a vertical axis. Inaddition to measuring these four reflectivities the four cases should also bemeasured with a non-magnetic scatterer such as graphite to enable the imperfectionof the flipper and polarizer efficiencies to be taken into account. A detaileddescription of neutron optics and magnetic effects can be found in ref [6,7].
Roughness and InterdiffusionThere is a large field of interest in what happens between materials at an inter-material boundary. An interface may be rough with peaks and troughs over a largerange of length scales with a fractal-like structure. It may also have magneticroughness if the magnetisation does not sharply change at an interface. A boundarymay be smooth but with one material diffused into the other. It turns out that in boththe rough and diffuse case the specular reflectivity is reduced by a factor very muchlike the Debye-Waller factor reduces scattered intensity from a crystal. Expression(12) would be affected in the following manner:
2.2-7
Figure 3: A schematic layout of a polarised neutron reflectivity experiment withanalysis. The initially unpolarised beam is polarised vertically by the polarisermirror. Flipper 1, when activated, inverses the polarisation allowing themeasurement of R+ and R-. The difference of these two data sets represents themagnetic contribution to the reflectivity. If in the process of reflection the neutronpolarisation was changed (flipped) due to a magnetisation component out of thevertical plane this can be detected by the use of an analyser mirror and secondflipper. Between the polariser and analyser a guide field (~20G) is required tomaintain the polarisation in the vertical direction.
(16)
where σ is a characteristic length scale of the layer imperfection. So what happens tothe intensity lost by the exponential factor in (16)? In the case of the diffuse interfacethe lost intensity must go into the transmitted beam as there are no potential gradientsin any other direction than normal to the surface. This is not the case for the roughinterface where intensity is lost by local reflections in directions away from thespecular direction or off-specular scattering. Information such as the height-heightcorrelation function can be deduced from this off-specular scattering [2,3].
Experimental TechniqueThe reflectivity can be measured as a function of q in two very different ways, eachof which has advantages and disadvantages. The first method known asmonochromatic involves making a θ-2θ scan with a well defined wavelength andstoring the reflected intensity for each θ. The incoming intensity must also bemeasured either at the same time with the use of a monitor and then scaled up bythe monitor efficiency or in the separate experiment. The reflectivity is simply theratio of these two intensities for each θ which is converted to q by Braggs law:
(17)
With a sample length of the order of a few cm and a starting θ of a fraction of adegree, the beam must be finely collimated to ensure the sample is underilluminated i.e. all the incoming beam strikes the reflection surface. The sample can
q = ( )4π ϑ λsin /
Rq
N ebqz≈
−16 2
42 2 2π σ
2.2-8
Figure 4: Two possible interfaces which can result in identical specularreflectivities. The data only differs in the case of the rough interface where off-specular scattering is observed.
be over illuminated for very small samples but the data must be corrected for thevarying flux on the sample as θ is increased.
As the reflectivity tends to drop quite fast with increasing q (see (12) whereR∝ 1/q4) in order to gain intensity (at the price of looser resolution) the collimation slitscan be opened keeping δθ/θ constant and equal to the fractional wavelength variation.The final curve in either case must have the resultant q resolution, δq, de-convolvedfrom the data. The q resolution is related to θ and λ by the following relation:
(18)
This method has the advantage that the wavelength chosen can be that with thehighest flux from the neutron source and therefore for a given resolution, is themost efficient method of using the flux available.
The alternative is to use time-of-flight (TOF). Here we keep θ constant anduse all the available wavelengths in the beam. The wavelength and hence q ismeasured by pulsing the incoming beam and measuring the arrival time to thedetector. The resolution is the same as (18) with δλ/λ replaced by δT/T where δT isthe pulse time width and T is the time-of -flight of the pulse. The resolution of thetime binning at the detector in principle is also a factor but in practice this is chosento be much smaller than δT. The range of q covered for a given θ depends on theuseful wavelength range which is a factor of 10 for the ILL reflectometer D17.Over this range however, the flux at the minimum and maximum wavelengthscompared to the peak flux may be more than 2 orders of magnitude smaller. Thehighest q (and lowest reflectivity) is measured by the shortest wavelength. For the
δ δθθ
δλλ
q
q
=
+
2 2 2
2.2-9
Figure 5: A typical collimation set-up within the reflection plane. When sample isunder-illuminated (minimal background), the distance between the last slit andthe sample should be as short as possible. In general the divergence out of theplane is defined only by the source.
same resolution the TOF method is less efficient than the monochromatic technique asit will always require longer counting times to measure the same q-range to the samestatistical accuracy. The reason the ILL reflectometer has both TOF andmonochromatic options is that kinetic experiments cannot be done in themonochromatic mode. If one had a sample which the layer structure was changing as afunction of time, only the TOF method can produce a unique R(q) for a given timerange. The θ-2θ measurement is a sequential set of counts so each point in q ismeasured at a different time. In order to solve the problem of reduced flux in TOF,D17 has a chopper system that can continuously vary the time resolution. Provided thereflectivity curve does not have fine structure then low a resolution measurement canbe completed in less than 1minute. The incoming beam intensity for TOF can bemeasured with a monitor in which case the data has to be re-binned in time beforedividing as the monitor is not at the same distance from the pulsed source as thedetector. In addition a separate measurement of the relative efficiencies of the detectorto monitor is required as a function of wavelength. On D17 there is no monitor for thisand the incoming beam is measured directly on the detector in a separatemeasurement. Care must be taken to avoid dead time losses where the local rate can bemuch higher than the mean count rate observed.
About the specular reflected beam in 2θ there is background that must besubtracted before dividing through by the direct beam. This background can besubstantial for experiments involving a solid-liquid interface where the substrate canbe pure water and limits the lowest reflectivity measurable to ~10-7. With a twodimensional multi-detector a wide range of 2θ is captured which will includebackground and off-specular diffraction. Instruments with a single detector require aspecial measurement with the detector in an off-specular position to measure thebackground. In both cases care must be taken in deciding the 2θ range for backgrounddoes not include diffracted intensity from the sample itself.
PNR experiments are complicated by the fact that the polarizers and flippers inthe system are not perfect. With no sample and incoming beam directly on the analyserwe would expect no flux on the detector in the condition -+ or +- for a perfect system.A measure of the efficiency of each flipper and the polarizers is the flipping ratio Fdefined as
(19)
where F1 and F2 are the flipping ratios for the first and second flippers. Theintensities are measured either with the direct beam for a single detector or with anon-magnetic scatterer for a multi-detector. The measured intensities, I correspondto the states of the two flippers as in figure 3. Acceptable flipping ratios are of the
F I I F I I1 2= =++ −+ ++ +− and
2.2-10
order of 40. These must be taken into account when attempting to extract themagnetic and nuclear components of the reflectivity [6,7].
ILL instrumentsAt present there are three reflectometers at the ILL, two of which are externallyfunded (ADAM & EVA) and therefore have limited time for ILL experiments. Theyare both fixed wavelength resolution (1%) monochromatic instruments with ADAM
2.2-11
Figure 6: The dual mode instrument D17 at the ILL designed to take advantageof both TOF and monochromatic methods of measuring reflectivity. (a) A sideview of the instrument in TOF mode. The monochromator is removed from thebeam and a double chopper system defines the time resolution (1-20% δδt/t).Between the collimation slits is a vertically focusing guide which increases theflux at the sample position at the price of increased vertical divergence of thebeam. The slits define the beam in the horizontal direction. (b) A vertical view ofthe instrument in monochromatic mode. Here the collimation arm is rotated (~4deg) to allow the beam reflected from the multi-layer monochromator system topass through to the sample. The choice of monochromators involves high and lowresolution (2-5%) plus polarising.
having a horizontal reflection plane and EVA vertical. EVA has been speciallydesigned to measure scattering from the evanescent wave mentioned above.
D17 is an ILL instrument with both monochromatic and TOF modes. Thewavelength resolution can be changed in the monochromatic mode and the timeresolution in TOF is continuously variable. All three have polarized neutrons withpolarization analysis. Unfortunately non of the three are capable of measuringreflection from a free liquid surface which requires a special instrument whichhopefully will be built in the near future at the ILL. Further details about theseinstruments can be found in the ILL web pages at http://www.ill.fr/YellowBook/.
References[1] Born and Wolf 'Principles of Optics', Permagon Press, Oxford (1970)
[2] Phys. Rev. B 'X-ray and Neutron scattering from rough surfaces' V38 N4 P2297
[3] J. Daillant A. Gibaud (Eds.) 'X-ray and Neutron Reflectometry: Principles andApplications', Springer (1999)
[4] V. F. Sears 'Neutron Optics', Oxford Press, Oxford (1989)
[5] X. Zhou S. Chen 'Theoretical foundation of X-ray and neutron reflectometry' Phys. Reports V257 p223 (1995)
[6] W. Williams 'Polarized Neutrons' , Oxford Press, Oxford (1989)
[7] A. Wildes 'Polarizer-analyser correction problem in neutron polarization analysis experiments' Rev. Sci. Inst. V70 p4241 (1999)
2.2-12
TIME-OF-FLIGHT NEUTRON DIFFRACTION
C. C Wilson
1. IntroductionThe principal characteristic of instruments for diffraction on pulsed neutron sourcesis their use of time-of-flight (TOF) techniques to allow for optimal use of the whiteneutron beams produced by the source. These rely on the fact that neutrons withdifferent energies, and hence different wavelengths, from de Broglie’s relationship,travel at different velocities:
λ=h/mv (1)
where m is the neutron mass and v the neutron velocity.Thus, since in the spallation process all neutrons in a given pulse are
essentially created at the same time, t0, the higher energy, shorter wavelengthneutrons travel faster and hence arrive at sample and subsequently at detector, at anearlier time than the lower energy, longer wavelength, slower neutrons. Bymeasuring the time of arrival of a neutron at the detector, and of course knowing itsflight path, we can calculate its velocity and hence its wavelength (energy). This isthe basis of TOF, which is a genuinely wavelength-sorted white beam technique ofgreat use in a wide range of diffraction techniques.
The wavelength and time-of-flight are related by the following expression:
λ= ht/mL (2)
where t is the time-of-flight and L the total flight path.And using Bragg’s law λ=2dsinθ, we get a relationship between TOF and d-
spacing:
t=m.L.2dsinθ/h (3)[=252.777.L.2dsinθ, with t in µs, L in m, d in Å]
We note here in passing that the commonly used unit for wavevector transfer inneutron scattering, Q=2π/d.
The measured d-spacing at a given scattering angle is therefore given by:
d=t/(252.777.L.2sinθ) (4)
i.e. the measured d-spacing is directly proportional to TOF. This has important
2.3-1
consequences for the design of TOF diffraction instruments. For example, if weconsider powder diffraction, Bragg’s law for a monochromatic instrument isnormally expressed in the form
λ=2dhklsinθhkl (5)
where λ is the chosen (fixed) wavelength and measurement of various Braggreflections hkl is achieved (at d-spacing dhkl) by scanning in angle (θhkl). However,in TOF techniques we have a range of wavelengths λhkl and we can rewrite theBragg equation as
λhkl=2dhklsinθ (6)
i.e. we can measure a whole range of d-spacings (Bragg reflections) at a fixedscattering angle 2θ by scanning in wavelength (≡TOF). This is the basic principleof TOF diffraction as a fixed-angle, wavelength-dispersive technique (Figure 1).
2.3-2
Figure 1: The basic principle of TOF diffraction. The time- (wavelength-) dis-persive white beam is incident on the sample, and the diffraction pattern record-ed at a fixed scattering angle 2θθ. The total flight path, L=L1 (source-sample dis-tance)+L2 (sample-detector distance). In this fixed geometry, the d-spacing dhkl isdirectly proportional to the time-of-flight (Eq. 4) and the diffraction pattern isrecorded as shown.
It is also, of course, possible to measure such TOF diffraction patternssimultaneously at many angles, in multi-detectors or in multiple detector banks.This can either lead to higher count-rates when the patterns measured in differentdetectors are summed and “focused” (usually in the data processing software), orcan lead to additional information being available in the diffraction patternmeasured in the different detector banks. This is a trend very much in evidence onmodern TOF diffractometers (Figure 2).
2. ResolutionAmong the important characteristics of a TOF diffraction instrument is that theresolution (∆d/d or ∆Q/Q) of a diffraction pattern measured at a given scatteringangle (or, for example, in a given “focused” detector bank)
(i) is constant over the whole diffraction pattern (the angular term ∆θcotθ inthe resolution being constant) and
2.3-3
Figure 2: An example of a modern multi-detector bank TOF diffractometer, theGEM instrument at ISIS. The detectors are arranged in a series of banks aroundthe sample tank, from low angles (lower left) to backscattering (upper right).GEM is used in the study of both crystalline (powder diffraction) and disorderedmaterials (liquids and amorphous scattering).
(ii) can be improved linearly with the length of the instrument (by minimisingthe term ∆L/L, where ∆L is a constant term reflecting the finite width of themoderator, and L is the length of the instrument).
As an example, the TOF powder diffractometer HRPD at the ISIS source has anoverall flight path of around 100m, and an effective moderator width of around 3cm(the moderator is 5cm thick but has a poisoning layer which reduces this). Thus the∆L/L term is of order 3 × 10−4. With the main detector bank in the backscatteringregion (2θ close to 180°, θ close to 90°), the angular term in the resolution, linear incotθ, is minimised and the overall resolution optimised. In the hi ghest resolution case,HRPD offers ∆d/d of around 4 × 10−4. Of particular note is that this very high ∆d/dresolution is maintained over the whole diffraction pattern (Figure 3).
3. Q-RangeAnother important aspect of a diffraction experiment is the Q-range (d-spacingrange) accessed in a single measurement. As can simply be seen, to get to lowvalues of d (high values of Q), one should measure at short TOF, i.e. with short
2.3-4
Figure 3: Powder diffraction pattern of Al2O3 measured on the high resolutiondiffractometer HRPD at ISIS. The excellent ∆∆d/d resolution is essentially con-stant over the whole pattern.
wavelength, high energy neutrons. Spallation sources are rich in such neutrons andso are well suited to short d/high Q measurements. For maximum Q-range it is alsoimportant to measure at a range of angles (in addition to measuring over a widewavelength range), as lower angles give generally lower Q data, while higherangles give access to higher Q.
4. Powder DiffractionThe principles of TOF powder diffraction have been discussed above. Highresolution diffractometers can allow the measurements of d-spacings of as low as0.25 Å and refinement of structures of up to around 100 atoms.
5. Single Crystal DiffractionSingle crystal diffraction on a TOF instrument is by far most efficiently carried outusing area detectors. White beam single crystal diffraction with an area detector (and astationary crystal) is termed Laue diffraction, while adding wavelength discrimination
2.3-5
1/λmin
1/λmax2θ
max
2θmin
2θmin
2θmax
Incident Beam
Figure 4: The geometry of TOF Laue diffraction. The two Ewald spheres aredefined by the minimum and maximum wavelengths used – they have radii1/λλmin and 1/λλmax. The 2θθmin and 2θθmax values are defined by the angles sub-tended by the area detector. In this two-dimensional projection, all areas ofreciprocal space between the Ewald sphere surfaces and the dotted lines areobserved in a single measurement with a stationary crystal and detector; thereciprocal lattice points coloured in black (along with the spaces in between).
using TOF techniques leads to the TOF Laue diffraction used on such instruments. Theparticular feature of this technique is that it allows access to large volumes ofreciprocal space simultaneously in a single measurement (Figure 4). These reciprocalspace volumes, moreover, are completely resolved in three-dimensions.
This means that a single measurement accesses many Bragg reflections butalso offers a wide survey of reciprocal space, ideally suited to the study of otherfeatures such as diffuse scattering which do not necessarily occur at the Braggreflection positions. Routine determination of organic and other structures of 100 ormore independent atoms is possible, with full anisotropic refinement of thermalparameters, and access to d-spacings of as low as 0.25 Å in simpler materials.
6. Diffraction From Disordered MaterialsThe important aspect of studies of materials which exhibit short range order, such asliquids, amorphous materials and highly disordered crystals, is to access a very wideQ-range in the measured pattern, S(Q). Only by so doing, along with very accuratedetermination of the instrument normalisation and background, can the pattern bereliably inverted (by Fourier transformation) to the pair distribution function g(r).
This g(r) function gives information about the relative distributions of theparticles (e.g atoms or molecules) in the sample. While the principles of the methodused for collecting TOF diffraction data from disordered materials are very similarto those for powder diffraction (Figure 1), in this case it is especially important touse a wide wavelength range at multiple angles (Figure 2; the GEM instrument isused to study both crystalline and disordered materials). However, it is also vital tokeep the detectors well calibrated (both in terms of neutron detection performanceand position) and very well shielded (to reduce background). Good secondarycollimation (shielding between the sample and detector) is therefore usuallyemployed, and the stationary detector arrangment is beneficial here. Q values of upto 50 Å −1 can be accessed.
7. Small Angle Neutron Scattering (SANS)As its name implies, SANS is a forward scattering technique. Once again thestrength of TOF SANS is in allowing simultaneous access to a very wide Q-range.By coupling the wavelength range allowed by the TOF technique with a fairly wideangle subtended by the area detector usually employed, TOF SANS typicallyaccesses Q ranges of ~0.005 – 0.2 Å −1 in a single measurement without moving thedetector. This has the benefit of allowing often complex 3D modelling to be carriedout on the scattering patterns.
2.3-6
8. Neutron ReflectivityOnce again NR on a TOF source is a fixed angle technique. It is usually carried outon a horizontal sample as shown in Figure 5.
The use of a fixed horizontal sample in this way greatly simplifies the study ofliquid surfaces.
The reflectivity patterns are recorded in total reflection geometry θi=θr, in thefixed detector (Figure 6), with the Q-range being provided by the wavelengthscanning of the TOF method.
Interference fringes are observed in the reflectivity profile if there is avariation in the scattering length density perpendicular to the surface (see Chapteron Reflectometry). The determined variation in scattering length density can reflectchanges in chemical composition, or the presence of, for example, a multilayerstructure. Detailed modelling of the scattering length density can be carried out,most reliably if a wide range of reflectivity (and Q) have been measured –reflectivities of as low as 10−6 can be accessed. Accurate modelling of thecomposition of adsorbed chemical systems is often achieved, as in SANS, by theuse of contrast variation. In this method selected components of the system aredeuterated, the large variation in scattering lengths between H and D leading tosubstantial changes in the scattered intensities, thus offering extra information toassist in reliable model building. An effective "resolution" of around an Ångstromin layer thickness is easily obtained in NR.
2.3-7
Figure 5: The geometry of TOF neutron reflectometry. The white beam (I) isincident upon a horizontal sample, and the reflectivity profile (R) is recorded asa function of TOF at a fixed angle, providing the IR(Q) used to analyse the char-acteristics of the sample perpendicular to the surface.
Off-specular reflection can also be measured, usually using an area detector, andgives information on, for example, surface roughness or on local surfaceinhomogeneities.
9. Sample EnvironmentIn common with all neutron scattering techniques, the use of sample environmentin TOF diffraction is a major strength. There are specific advantages inherent in theTOF technique, however, related to the ability to measure the entire diffractionpattern at a single, fixed scattering angle. For complicated, bulky sampleenvironment apparatus, it may therefore be possible to leave only small apertures,for the incoming beam and at limited fixed scattering angles, yet still measure thediffraction of interest (Figure 7).
Typical applications of this method include high pressure cells and thick-walled chemical reaction cells. The high degree of collimation means thatextraneous scattering from these bulky items is minimised, and the signal from the(sometimes small) sample optimised.
10. Neutron Strain-ScanningThe neutron strain-scanning technique basically involves measuring diffraction
2.3-8
Sample
S1
S2S3
S4
Mo
nitor
1
Fra
me
Overlap
Mir
rors
Superm
irro
r
Mo
nitor
2
Dete
cto
r
Figure 6: Schematic view of the CRISP reflectometer at ISIS, showing the hori-zontal geometry, the arrangement of apertures, the horizontal sample and thefixed detector.
patterns from limited gauge volumes within a bulky engineering component, andcharacterises internal stresses and strains from the changes in position and shape ofthe measured Bragg peaks. The benefits of TOF for neutron strain-scanning followfrom the above discussions of TOF powder diffraction and sample environment.
11. Further informationFurther information on time-of-flight diffraction, instrumentation, techniques andapplications can be found on the WWW site of the ISIS Pulsed Neutron and MuonFacility, Rutherford Appleton Laboratory, UK (http://www.isis.rl.ac.uk).
2.3-9
Figure 7: The TOF neutron diffraction technique is well suited to studying sam-ples inside bulky sample environment apparatus, the ability to measure thewhole diffraction pattern at a single scattering angle (Eq. 6; Figure 1) being animportant factor in this. Here the sample is contained inside apparatus with onlynarrow apertures for incoming and outgoing beams and at very limited scatter-ing angles necessary.
2 THE FUNDAMENTAL EQUATIONS2.1 The nuclear structure factorFor a non-Bravais single crystal, i.e. a crystal with more than one atom per unit cell,the nuclear structure factor at the reciprocal lattice point reads [1]:
where and Wj are the Fermi length and the Debye-Waller factor of the atom jlocated at the position of the unit cell.
2.2 The magnetic interaction vectorFor a non-Bravais single crystal, the magnetic interaction vector at the reciprocallattice point is given by [1]:
where:
10-12 cm/µB,
is the form factor and is the Fouriemponent of the magnetic moment ofthe atom j. One must notice that because of the dipolar nature of the magneticinteraction, the neutron is only sensitive to the projection of on a planeperpendicular to the scattering vector .
2.3 The scattering cross-section and the scattered polarisationUsing the density matrix formalism, Maleyev et al. [2,3] and Blume [4] couldpredict the scattering cross-section and the scattered polarisation vector within the Born approximation. Both fundamental equations are the sum of fourparts summarised in table 1 [8], where and are the incident and final wavevectors of the neutron beam, and is the Van Hove correlation function:
±± = ±±
rv vMQ Q⊥ −( ),, NN
Q Q− ⊥( )v v
r,
,
MN− ⊥( )v v
rQ Q
M,,
A B
k T
A B
b
, ,( ) =− −( )( )
″
11π ω ω
exp h
rk f
rki
rPσ t
rQ
r rM Q( )
rk
r r
mkj
f j
rQ( )
r r r r r rr
M Q Q m Q rk( ) = ( ) ( ) −( )∑ •f i Wjj
j j jexp exp .γ 2e
20 26952mec
= .
r r r r r rM Q Q M Q Q
⊥ ( ) = ∧ ( ) ∧γ 2
22
em ce
rQ
rrj
b j
−
N b i Wj jj
r rQ Q r( ) = ( ) −( )•∑ j exp exp
rQ
2.4-2
rQ 2
3 THE POLARISED NEUTRON TECHNIQUES3.1 The spin-dependent diffraction techniqueThe polarised neutron diffraction technique takes advantage of the incidentpolarisation dependency of the cross-section to measure precise quantitativemagnetisation distributions of single crystals. Its sensitivity to the distribution ofunpaired electrons in the unit cell is greater than for the conventional method usingunpolarised-beams (about 0.1 mµB). This technique is mainly used for investigatingsingle crystals that are ferro- or ferri-magnetically ordered in an applied magneticfield. It can also be applied to some antiferromagnetic materials where
or
If the beam is perfectly polarised parallel (+) or opposite (-) to the applied field (Pi
= ±1), and if the sample is ferro- or para-magnetic and has a centre of symmetry, weget the simple expression for the flipping ratio:
where α is the angle between the polarisation vector (i.e. the applied field) and thedirection of the scattering vector . From experimental values of R( ), the
rQ
rQ
r rM M⊥ ⊥∧ ≠* 0N *
rM( ) ≠ 0
σ t
2.4-3
Table 1: Scattering cross-section and scattered polarisation
vector .
σ π σt jjN
v = ( ) ∑0
3
0
2
RN M
N Mt
t
rr r
r rQQ Q
Q Q( ) = =
( ) + ( ) ( )( ) − ( ) ( )
+
−
σσ
α
α
sin
sin
2
r rP N
vP
tj
jσ π σ = ( ) ∑ 0
3
0
2
scattering. In order to separate each of these components it is necessary to usepolarised neutron techniques. For example, in order to study spin correlations andmagnetic defect structures in systems such as paramagnets, spin glasses andantiferromagnets, so called XYZ-polarisation analysis (XYZ–PA) is required [7].
In XYZ–PA, the longitudinal component of polarization of the scattered beamalong the arbitrarily defined x- y- and z-directions is measured. Using thistechnique, and ensuring that the scattering vector is always in the x-y plane (definedwith respect to the neutron polarization vector), one can separate the nuclear,magnetic and spin-incoherent structure factors unambiguously, on a multi-detectorspectrometer, by combining the six measured cross-sections (x-, y- and z-directions, spin-flip and non-spin-flip) given below:
The subscripts nc, mag, si, and ii stand for “nuclear coherent”, “magnetic”, “spinincoherent” and “isotope incoherent” contributions respectively. These equationsapply to systems with collinear magnetisation and a randomly oriented momentdirection (i.e., a paramagnetic system or an antiferromagnetic powder in zeroexternal field). In antiferromagnetically ordered single crystals, there will, ingeneral, be a strong correlation between the x and y components of the samplemagnetisation , i.e. the components of in the scattering plane. Therefore, inorder to separate the magnetic cross-section in an ordered antiferromagnetic singlecrystal, either the angle between and , or the magnetic structure factor of thesample must be known in advance. In practice, if neither of these quantities areknown, then application of the equations will result merely in a possible change ofsign of the magnetic intensity, depending on the moment direction. In the case ofnon-collinear systems, such as helical magnets, cross-terms appear in the magneticinteraction potential and the equations do not hold [2–4]. The XYZ–PA methodcannot be applied to the study of ferromagnets since ferromagnetic domains anddemagnetisation fields will in general depolarise the neutron beam.
3.4 Zero-field spherical neutron polarimetryThe only way to measure the three components of the scattered polarisation is toconnect two independent magnetic guide-fields onto a zero-field sample chamber.Cryopad is presently the only device able to achieve full vectorial control of theneutron polarisation over a large range with a good precision [8]. The sample
rQ
rQ
vM
vM
vM
σ α σ σ σ σ
σ α σ σ σ σ
σ σ σ σ σ
xnsf
mag si nc ii
ynsf
mag si nc ii
ynsf
mag si nc ii
= +( ) + + +
= +( ) + + +
= + + +
12
113
12
113
12
13
2
2
sin
cos
σ α σ σ
σ α σ σ
σ σ σ
xsf
mag si
ysf
mag si
zsf
mag si
= +( ) +
= +( ) +
= +
12
123
12
123
12
23
2
2
cos
sin
2.4-5
σ znsf
chamber is maintained in a zero-field state, preventing any polarisation precessiondue to a parasitic field.
In the case of magnetic structures, spherical neutron polarimetry (SNP) allowsthe direct determination of the direction and phase of the magnetic interactionvector, and for antiferromagnetic structures, SNP is a powerful method formeasuring precision form factors and magnetisation distributions [9].
Because transverse components contain information related to thenuclear–magnetic interference terms, SNP is clearly the best method to studyantiferromagnetic compounds for which the magnetic and nuclear information arelocated at the same place in the reciprocal space. In the inelastic case, one should beable to observe the spin–lattice correlations expected in geometrically frustratedantiferromagnets or spin-Peierls systems like CuGe03.
Here are a few rules describing the polarisation behaviour when carrying outSNP experiments:– The polarisation is conserved when the interaction is purely nuclear and coherent(e.g. an isolated phonon or a nuclear Bragg peak without nuclear spin polarisation).– The polarisation precesses by around when the interaction is purelymagnetic non-chiral (e.g. a magnetic satellite of a collinear arrangement or amagnon).
– In the case of a non-collinear magnetic structure, the intensity may depend on
and for an helix, may create polarisation along the scattering vector .
– When there are nuclear–magnetic interferences, the intensity depends on when
N and are in phase. Polarisation is created along when N and are in
phase (e.g. Fe2O3), and the polarisation rotates around when N and are in
quadrature (e.g. Cr2O3).
References1. G.L. Squires, “Thermal neutron scattering”, Cambridge University Press.
2. S.V. Maleyev, Physica B 276-268 (1998) 236
3. S.V. Maleyev, V.G. Bar’yakhtar, R.A. Suris, Fiz. Tv. Tela 4 (1962) 3461 [Sov.Phys. – Solid state 4(12) (1963) 2533]
4. M. Blume, Phys. Rev. 130 (1963) 1670
5. G.M. Drabkin et al., Sov. Phys. JETP 20 (1965) 1548
6. R.M. Moon, T. Riste and W. Koehler, Phys. Rev. 181 (1969) 2533
7. O. Schärpf and H. Capellmann, Phys. Stat. Sol. A135 (1993) 359; O. Schärpf and
rM⊥
rM⊥
rM⊥
rM⊥
rM⊥
rPi
rQ
r rM M⊥ ⊥∧*
rPi
rM⊥
rτ = 0
2.4-6
H Capellmann, Z. Phys. B - Condens. Matter 80 (1990) 253
8. F. Tasset, Physica B 156-157 (1989) 627; V. Nunez, P.J. Brown, J.B. Forsyth andF. Tasset: Physica B 174 (1991) 60; P.J. Brown, J.B. Forsyth and F. Tasset: Proc. R.Soc. Lond. A 442 (1993) 147; F. Tasset, Physica B 297 (2001) 1; P.J. Brown,Physica B 297 (2001) 198
9. P.J. Brown, J.B. Forsyth and F. Tasset: Physica B 267-268 (1999) 215; P.J.Brown, J.B. Forsyth, E. Lelièvre-Berna and F. Tasset: J. Phys.: Condens. Matter 14(2002) 1
2.4-7
Magnetic Form FactorsP.J. Brown
The magnetic form factor ƒ(q) is obtained from the fourier transform of themagnetisation distribution of a single magnetic atom. Assuming that it has a uniquemagnetisation direction it can be written
1)
where M gives the magnitude and direction of the moment and m(r) is a normalisedscalar function which describes how the intensity ofmagnetisationvaries over the vol-ume of the atom. When the magnetisation arises from electrons in a single open shellthe magnetic form factor can be calculated from the radial distribution of the electronsin that shell. The integrals from which the form factors are obtained have the form
2)
The jl are the spherical Bessel functions defined by\begin
3)
If the open shell has orbital quantum number l the form factorfor spin moment is
4)
and that for orbital moment
5)
The coefficients SLM, BLMhave to be computed from the orbitalwave-function [1].The total spin moment Ms is given by S00 and the orbital moment ML by B00. Forsmall q the dipole approximation
6)Tables 1-4 give coefficients of an analytic approximation to the ‹j0› integrals for thed electrons in ions of the 3d and 4d series, and the f electrons of some rare earth andactinide ions. The approximation has the form
7)For these expansions s = sin θ/λ in units of Å-1.The integrals ‹jl› with L ≠ 0 are zero when s = 0 and have been fitted with the form
8)
2.5-1
Tables 5-13 give the coefficients obtained. For the transition metal series the fits weremade withform factor integrals calculated from Hartree-Fock wave-functions [2]. Forthe rare-earth and actinide series the fits were with Dirac-Fock form factors [3, 4].In the tables the number following the atom symbol indicates the ionisation state ofthe atom. Thus the coefficients following Fe0 are for a neutral iron atom and thosefollowing Fe2 are for Fe2+
References[1] Marshall W and Lovesey S W (1971) Theory of thermal neutron scattering,Chapter 6, Oxford University Press.[2] Clementi E and Roetti C (1974) Atomic Data and Nuclear Data Tables 14, 177-478.[3] Freeman A J and Descleaux J P (1979) J. Magn. Mag. Mater. 12, 11-21.[4] Descleaux J P and Freeman A J (1978) J. Magn. Mag. Mater. 8, 119-129.
2.5-2
2.5-3
2.5-4
2.5-5
2.5-6
2.5-7
2.5-8
2.5-9
2.5-10
2.5-11
2.5-12
Time-of-Flight Inelastic NeutronScattering
R. S. Eccleston
IntroductionIn the inelastic neutron scattering experiment, the quantity we measure is thedouble differential cross section:
The velocity of thermal neutrons is of the order of km s-1; consequently their energycan be determined by measuring their time-of-flight over a distance of a fewmetres. In the inelastic neutron experiment the parameters of interest are the energyand momentum transfer, hω and Q respectively. Time-of-flight spectrometers may be divided into two classes:-• Direct geometry spectrometers: in which the incident energy, Ei, is defined by
a device such as a crystal or a chopper, and the final energy, Ef, is determined bytime-of-flight.
• Indirect (inverted) geometry spectrometers: in which the sample isilluminated by a white incident beam and Ef is defined by a crystal or a filterand Ei is determined by time-of-flight.
At a pulsed source all spectrometers use the time-of-flight techniques. On steadystate sources pulsing devices such as choppers are required.
( )ωσ
,22
Qk
kSb
dEd
d
i
f=
Ω
2.6-1
Figure 1: Distance-time plot for a direct geometry spectrometer (left) and anindirect geometry spectrometer (right).
The mode of operation of each type of spectrometer can be clearly understoodif the distance-time plots are considered (figure 1).
Accessible regions of (Q,ωω) spaceTo understand the trajectories traced out through (Q,ωω), consider the scatteringtriangles. Clearly for direct geometry spectrometers ki is fixed and kf varies as afunction of time whereas the reverse is true for indirect geometry instruments.
Applying the cosine rule
For direct geometry Ef can be eliminated leaving
Each detector has a parabolic trajectory through (Q,ω) space.
To make optimum use of direct geometry spectrometers, they are equipped withlarge detector arrays, giving simultaneous access to a large area of (Q,ω) space.
Similarly for indirect geometry Ei can be eliminated giving
The parabola are inverted. An important feature of the indirect geometry instrumentis the access to a wide range of energy transfers for energy loss.
( )[ ] 21
22
cos222
ωφω hhh
+−+= fff EEEm
Q
( )[ ] 21
22
cos222
ωφω hhh
−−−= iii EEEm
Q
( ) φcos22
21
22
fifi EEEEm
Q −+=hφcos2222fifi kkkkQ −+=
2.6-2
Figure 2: Scattering triangles for a direct geometry spectrometer (left) and anindirect geometry spectrometer (right).
Generic instrumentsThe table (table 1) provides muchsimplified examples of direct andindirect geometry instruments. Thecritical differences between the two arethe manner in which they cover (Q,ω)space and their resolution characteristics.On a steady state source, the moderatoras it is shown in the table is replaced by apulsing device such as a chopper.
The chopper spectrometerThe chopper is a ubiquitous feature of direct geometry spectrometers for reasonsgiven later in the chapter. Examples of chopper spectrometers include HET, MARIand MAPS at ISIS and IN4, IN5 and IN6 at the ILL. In the case of some time-of-flight spectrometers on steady state sources the chopper simply provides the pulsedstructure of the beam and a crystal monochromator is used to select the incidentenergy, for example IN6. Other instruments such as IN5 use an array of choppersboth to provide the pulsed structure and to monochromate the beam. On a pulsed source the design of a chopper spectrometer is in general quitestraightforward, with the key components arranged as indicated in the table. The
2.6-3
Figure 3: Trajectories through (Q,ωω) space: a direct geometry spectrometer(left) and an indirect geometry spectrometer (right) for detector at the givenscattering angles.
Table 1: Generic instrument types.
beam is monochromated using a Fermi chopper. A second chopper is usuallyinstalled effectively to close the beamline at the time at which the proton beam hitsthe spallation target, thereby preventing a large flux of epithermal neutrons enteringthe spectrometer where they will thermalise and produce a background signal. Inall cases detector arrays tend to be as large as economically and physically possibleto maximise the efficiency of the spectrometer.
ResolutionFor the sake of simplicity the resolution function of a chopper spectrometer isconsidered. The energy resolution has two contributions: from the moderator pulsewidth and from the band width of the chopper pulse. Concisely the expression canbe written:
.
Where ∆tc is the chopper burst time width, tc is the time of flight at the chopper and∆tm is the moderator pulse width. L1 is the moderator to chopper distance, L2 thesample to detector distance and L3 the chopper to sample distance.
Single crystal experiments on a chopper spectrometerDirect geometry spectrometers have, in the past, been thought of as being well adaptedfor studying excitations in non-crystalline or polycrystalline samples, but not suitable forsingle crystals. However, the development of instrumentation and experimental tech-niques has shown that direct geometry spectrometers can be used very effectively forsuch studies allowing a broad survey of (Q,ω) space but also detailed, focussed studies.
From the scattering triangle we can see that an array of detectors will trace outa sector in reciprocal space, and as we have seen earlier, each detector has aparabolic trajectory through (Q,ω) space. Consequently a broad detector array
produces a surface in (Q||,Q⊥,ω) space.Consider the example of a ferro-
magnetic spin wave in a three-dimensionalmagnetic system. As the dispersion intersectsthe surface traced out by the time-of-flight
∆ ∆ ∆h h hω ω ωE
t
t
L L
L E
t
t
L
L Ei
c
c i
m
c i
= + + −
+ + −
21 1 2 1 11 3
2
32
2
3
2
32
21
2
2.6-4
Figure 4. Use of a multidetector tocover a region in (Q, Q⊥⊥) space.
trajectories for the detectors, scattering will be observed. The development of large position sensitive detector (PSD) arrays provides
greater flexibility for single crystal measurements. The MAPS spectrometer at theISIS pulsed source has a detector area of 16m2 divided up into approximately50000 pixels. For any given crystal orientation arbitrary cuts can be made acrossthe surface of the detector, providing the ability to collect data along severalcrystallographic directions simultaneously. In essence, the spectrometer collects alarge (0.5Gbytes) volume of data and the choice of scans is made in software afterthe experiment. For more information see reference (Perring).
Indirect geometry spectrometersIt is apparent from figure 2 that indirect geometry spectrometers offer access to awide range of neutron energy loss values. They also offer high resolution at theelastic line and tend to offer broad coverage in terms of energy transfer withreasonable resolution.A variety of spectrometer designs offer different capabilities:
• Crystal analyser spectrometer : molecular spectroscopy• Back scattering spectrometer : high resolution• Coherent excitations spectrometer : coherent excitations in single crystals.
Before discussing each of these in turn, it is useful briefly to consider the resolutioncharacteristics of an indirect geometry spectrometer.
The resolution of indirect geometry spectrometersFor the indirect geometry spectrometer the energy resolution contains termspertaining to the uncertainty in the angular spread of neutrons scattered from theanalyser, ∆θA and timing errors over the incident flight path, ∆t. Timing errors arisefrom several contributions, including moderator thickness, sample size broadening,analyser thickness broadening, detector thickness and the finite width of datacollection time channels. If ∆t is expressed in terms of an equivalent distanceδ=hκi∆t/m the energy resolution can be concisely expressed as:
where L1 is the moderator to sample distance, and L2 the sample to detectordistance. Clearly increasing θA or L1 improves resolution but physical andinstrumental limitations apply.
21
2
23
1
22
1
1cot2
+∆+
=
∆
f
iAA
f
i
i E
E
L
L
E
E
LEθθ
δωh
2.6-5
Back-scattering spectrometerFor a crystal analyser spectrometer, the resolution contains a cotθA term which canbe reduced to almost zero by using a back-scattering geometry, thereby optimisingthe definition of Ef. In a matched spectrometer, Ei should be determined to the sameprecision, hence L1 tends to be long. IRIS at the ISIS Facility represents anexcellent example of such an instrument. In the most frequently used mode ofoperation the 002 reflection from graphite analysers defines an Ef of 1.845 meVand provides resolution of 17µeV. The analysers are cooled to reduce thermaldiffuse scattering (TDS), which broadens the elastic peak and gives rise toadditional background.
Crystal analyser spectrometers For molecular spectroscopy, energy information is often more important than Qinformation. Consequently, measuring energy transfer along a single trajectoryprovides an efficient method for measuring excitations across a broad spectralrange. TOSCA at ISIS provides a good example. A graphite analyser defines an Ef
of 3meV, a cooled beryllium filter is used to remove higher order reflections. Thegeometry of the secondary spectrometer is such that the sample, analyser anddetectors are parallel; this effectively time-focuses the scattered neutrons, reducinguncertainty in the scattered neutron flight time.
Coherent excitations - PRISMAOne of the shortcomings of chopper spectrometers and the direct geometryspectrometers described above is their inability to perform scans uniquely along
2.6-6
Figure 5: Schematic of the PRISMAspectrometer.
Figure 6: Time-of-flight scan withthe PRISMA spectrometer.
high symmetry directions. By adopting a scattering geometry whereby thecondition sinφ / sinθA = constant is met (figure 5), time-of-flight scan scans for alldetectors correspond to scans along a defined direction in Q space (figure 6).
The PRISMA spectrometer can operate in this mode. It is technicallychallenging because the movement of analysers is limited by the risk of clashing.
Monochromating devices ChoppersAs mentioned above, choppers are key components of a direct geometryspectrometer. For instruments using neutrons in the thermal to high energy ranges,Fermi choppers are most appropriate; for cold neutron instruments disk choppersoffer inherent advantages.
Fermi choppers A Fermi chopper is effectively a drum with a hole through the middle. The hole isfilled with alternating sheets of neutron absorbing material (slats) and transparentmaterial (slits). The slits and slats are curved, with the radius of curvature and theslit/slat ratio optimised for specific energy ranges. On the ISIS chopperspectrometers for example three slit packages are used to cover the incident energyrange from 15meV to 2eV. An additional slit package with very broad slits is usedto improve intensity by degrading resolution. The Fermi chopper can be rotated atfrequencies of up to 600Hz. The incident energy is defined by the phase of thechopper relative to the incident neutron pulse. The resolution is dictated by theslit/slat ratio of the slit package and the chopper frequency. Fermi choppers allowcontinuous selection of Ei.
Disk choppersDisk choppers are simply rotating disks with holes in them. They offer highertransmission than Fermi choppers with the same flexibility in the selection of Ei.Using two disks or a variable aperture chopper overcomes the need to change slitpackage. They are also compact. However their frequency is limited by engineeringconstraints, as is the thickness of the disks. Both these factors mean that diskchoppers are often not practical for thermal to high-energy neutron instruments.
CrystalThe use of crystal monochromators for time-of-flight spectroscopy is limited,because they impose some geometrical constraints, and reflectivity falls as energyrises. The geometrical constraints can be lifted by using a double bounce
2.6-7
arrangement, but at the expense of additional losses arising from the secondreflection. Focussing of the monochromator does offer the opportunity to traderesolution for additional flux.
As discussed above, the use of crystal analysers is widespread on indirectgeometry spectrometers.
FiltersThe Be filters effectively remove neutrons with an energy greater than the Braggcut-off of 5.2meV. Incident neutrons are scattered in the beryllium and thenabsorbed in absorbing sheets within the filter. They are used to remove higher ordercontamination in indirect geometry instruments. Their efficiency can be improvedby cooling, which reduces transmission of neutrons with energies above the cut-offthat arise from up-scattering by thermally excited phonons.
Nuclear resonant absorption from a foil is used for very high-energy measurements.The eVS spectrometer at ISIS, for example, uses a filter difference technique usinguranium or gold foils to measure atomic momentum distributions in the eV range.
Reference“Neutron scattering (mostly) from low dimensional magnetic systems” by T.G.Perring, in Frontiers of Neutron Scattering, edited by A. Furrer, (World Scientific,2000)
2.6-8
Three-axis spectroscopy
R. Currat and J. Kulda
Principle of the techniqueIn order to obtain information on the excitation spectrum at a given point Q in recip-rocal space, a more sophisticated procedure has to be adopted than just scanning theenergy of the scattered neutrons. At each point of the scan the scattering triangle ismodified, as indicated in Fig. 1, so that ki and kf will close at the same momentumtransfer Q but the length of one of the wave vectors – preferably ki – is varied toprovide for the required energy transfer. As is clear from the Fig. 1, in addition tothe ki and kf modification, also all the angles of the scattering triangle will change.Herein lies the price to be paid for the increased flexibility required for scans withQ = const.: the measurements have to be performed in a step-by-step mode, mak-ing multichannel data acquisition quite difficult.
2.7-1
Figure 1: Scattering geometry for a Q = const. scan.
Compared to a direct-geometry TOF instrument, this disadvantage is, howev-er, largely compensated by the high incident monochromatic flux due to the steady-state operation of the neutron source as opposed to a pulsed beam providing neu-trons for only about 1/1000 of the total measurement time.
As a general rule, the TAS is the technique of choice whenever information onexcitations in single crystals is sought [1] at a well defined point or along a givendirection in the Q,ω space and when a detailed quantitative interpretation of themeasured spectra is intended. On the contrary, if a global overview of the excita-tions in a broad Q,ω range is the goal, the TOF may be of advantage as soon as thenumber of useful detector channels can compensate the initial loss due to the timemodulation of the incident beam with a low duty-cycle.
The neutron three-axis spectrometer (TAS)
2.7-2
Figure 2: Schematic of a three-axis spectrometer.
A typical set-up of a three-axis spectrometer (TAS) is schematically displayedin Fig. 2. The incident and scattered neutron wave vectors, ki and kf, are selected byBragg diffraction on the monochromator and analyser crystals, respectively.Traditionally, large crystals of Cu, Zn or pyrolitic graphite (PG) with mosaic widthof 20 – 30 minutes have been used in combination with Soller collimators definingthe beam divergence of the incident and diffracted beams. On present instrumentsthe mosaic crystals are segmented into plates of a few centimeters size and mount-ed on mechanical devices (benders), which permit to control their individual orien-tations in order to act as curved mirrors and to focus the neutron beam horizontal-ly and/or vertically on the sample and detector [2]. More recently, elastically bentperfect Si and Ge crystals, offering improved focusing properties thanks to theabsence of random mosaic block misorientations, get employed for work requiringhigher resolution [3]. The monochromatic neutron flux incident on the sample ismonitored by a low-efficiency (≈ 10-5 - 10-4) detector (“monitor 1”), another sim-ilar “monitor 2” is placed in the scattered beam to detect the eventual presence ofstrong Bragg diffraction peaks, which might give rise to spurious signals in theinelastic spectra. The detector is usually a single 3He proportional gas tube.
The monochromator and analyser crystals diffract together with the nominalwavelength λ also its harmonics λ/2, λ/3 etc., which may be at the origin of spuri-ous effects. Usually, the most dangerous one is λ/2, because it is most intense andits energy is closest to the nominal one. Harmonic contaminations may be sup-pressed by using appropriate filters. For hot neutrons absorption filters are avail-able, using sharp absorption resonances of certain nuclei in the 0.1 - 1 eV range. Forthermal neutrons pyrolithic graphite (PG) permits to suppress efficiently the secondorder for two "magic" wavelengths of λ = 1.53 Å and 2.36 Å, corresponding to k= 4.1 Å–1 and 2.66 Å–1, respectively. For cold neutrons cooled polycrystallineberyllium permits to cut all wavelengths below 4 Å, ie. to transmit only neutronswith k < 1.57 Å-1. Alternatively, silicon and germanium crystals, whose odd-hklreflections are free from 2nd order contamination, can be used for certain applica-tions.
The TAS resolution function The resolution volume, i.e. the region of the Q,ω space which corresponds to themomentum and energy transfers of neutrons scattered and registered in a givenspectrometer configuration, can be approximated by a rather anisotropic 4D ellip-soid. Its form is determined by the convolution product of the two reciprocal spacedistribution functions pi(ki) and pf(kf) which describe the transmission of the mono-chromator and analyser arms [4]. For a spectrometer configuration characterised by
2.7-3
nominal values Q0,ω0 of Q,ω the measured intensity is given by
I(Q0,ω0) = N A(kI) ∫ R(Q-Q0,ω−ω0) S(Q,ω) dQ dω, (1)
where A(kI) and S(Q,ω) describe the source spectrum and the sample scatteringfunction, respectively. The norm of the resolution function is the product of thenorms of the two distributions:
∫ R(Q-Q0,ω−ω0) dQ dω = ∫ pi(ki) pf(kf) dki d kf = VI VF . (2)
In the gaussian approximation [4], [5] the TAS resolution function depends on thekinematic variables ki, kf, which define the energy and momentum transfer Q, ωand on the collimation angles, mosaic widths and scattering senses. For a flat mosa-ic monochromator crystal (i.e. in the absence of focusing) one gets:
, (3)
where kI is the average incident neutron wave vector, kI =VI-1 ∫ ki pi(ki) dki, and
α0, β0, α1 and β1 are respectively, the horizontal and vertical beam collimationsbefore and after the monochromator; ηm and η’m are the horizontal and verticalmosaic widths of the monochromating crystal and Pm(kI) its the peak reflectivity.The expression for VF is analogous.
Typical beam divergences are about 0.5 – 2 degrees in both horizontal andvertical directions leading to relative momentum transfer resolution ∆Q/Q ≈ 10-2.The overall energy resolution lies usually in the range of 5 – 10%, but can beimproved in particular cases. Depending on the spectrum of the neutron source –hot, thermal or cold – the TAS can be employed for studies of excitations in theenergy range from 50 µeV to 200 meV.
In reality the resolution volume has a more complicated shape, influenced byspatial variables (source size, slit widths, sample dimensions) and by the eventualuse of composite crystals made up from platelike segments to obtain approximatehorizontal and/or vertical focusing. Originally, the Gaussian approximation hasbeen extended to enable a more realistic description of the TAS resolution function[6]. More recently, the RESTRAX [7] and McStas [8] computer packages becameavailable, which provide a highly realistic Monte-Carlo ray-tracing simulation ofthe whole instruments including effects of neutron guides, slits etc. whose intrinsictransmission profiles are not necessarily represented by gaussian distributions.
V PI m I I M= ( ) ( )k k
4 + + 4sin ( ) ' + + m
m2
02
12
0 1
2m m
20
21
2
3 3
0 1
2cot
. .
θ πη α α
η α αβ β
θ η β β
2.7-4
Preparation and optimization of a TAS experimentThe first thing to be made in preparing a TAS experiment is to define the requiredneutron energy range matching the problem to be studied. Next a suitable instru-ment installed at a corresponding neutron source (cold, thermal or hot) is to beselected. Among the main factors influencing such a decision we may note
• the kinematic constraints, corresponding to the momentum conservation law(“closing the triangle”), to be satisfied over all the required Q, ω range; theseconstraints are particularly severe in studies of magnetic materials, where low-Q and large-ω often appear in combination
• the resolution/intensity conditions required for a given experiment; anyimprovement in resolution is paid for by a loss in countrate
• the angular limitations on the chosen instrument, notably the maximum valueof the scattering angle q, which determines the maximum value of Q that maybe reached for a given set of ki, kf values
• the risk of spurious signals; as a rule of thumb the final energy should be com-parable to the energy transfer in the downscattering mode in order to minimiseharmonic contaminations (see below) and the same applies to the incidentenergy in the upscattering mode Table 1 indicates typical neutron energies and instrument parameters for a
quick overview. Precise values depend on the detailed characteristics of each instru-ment, the fluxes correspond to the instruments at the High Flux Reactor at ILLGrenoble (France). The flux values available at smaller reactors may be 1-2 ordersof magnitude lower. Optimising the experimental conditions also involves makingchoices and compromises with respect to the d-spacing of the monochromator and
2.7-5
Table 1: Typical neutron energies and instrument parameters for TAS spec-trometers at the ILL.
TAS cold thermal hot
Mono/analyser PG002 PG002 Cu220
Flux [107 cm-2s-1] 1 - 3 3 - 30 1 - 3
ki [Å-1] 1 – 2.66 2.66 - 6 5 - 15
Qmax [Å-1] 2 6 12
∆Q [Å-1] 0.003 0.01 0.03
E [meV] 0.1 - 10 5 - 60 50 - 200
∆E [meV] 0.05 – 0.5 0.8 - 4 4 - 10
analyser crystals, collimations, the use of flat or focusing bent crystals, the scatter-ing configurations. When carrying out this complex optimisation process oneshould use one of the simulation programs mentioned above.
Data treatment.As a rule, the intensities measured in a constant-Q scan should be corrected pointby point for the variation of the norm of the resolution function (zeroth-order reso-lution correction).
For a scan obtained in the constant-kf mode, VF is constant across the scan and themeasured intensities should be corrected for the variation of A(kI) VI. Since the quan-tity A(kI) kI VI measures the neutron flux incident on the sample, it is sufficient to usea monitor with a 1/vn characteristic in the incident beam to normalise the counting timeper point. However the presence of higher-order harmonics (λ/2, λ/3, …) in the inci-dent beam requires, in general, additional corrections to be applied (see below).
For a scan obtained in the constant-ki mode the measured intensities must becorrected for the variation of Pa(kF) kF
3cotθa. Note that Pa(kF) may vary rapidly asa function of kF due to parasitic multiple Bragg reflections.
Spurious signals.An element, which may perturb the measurements, is the presence of the harmonicwavevectors 2ki, 3ki, … in the primary beam or 2kf, 3kf, ... in the scattered beam dueto higher-order diffraction at the monochromator or analyser position. For instance,the contribution of the unfiltered 2ki and 3ki harmonics to the monitor count-rate(used to normalise the acquisition times in the constant-kf mode) can be quite sig-nificant and lead to large scan profile distorsions unless corrected for.Harmonics also lead to spurious signals for “commensurate” values of ki and kf:
nki = mkf with n,m = 1,2,3,4,..due to elastic (coherent or incoherent) scattering on the sample (or its environment). To minimise the probability of such processes one should limit the magnitude of theenergy transfer with respect to the incident neutron energy. A safe rule of thumb isgiven by 2/3 < ki/kf < 3/2.
Spurious “inelastic” peaks are observed for 3-axis configurations such that 2crystals out of 3 (sample + monochromator or sample + analyser) satisfy a Braggcondition for a given neutron energy. These configurations are illustrated schemat-ically in Figs. 3a and 3b.
The spurious process shown in Fig. 3a (“type I”) can be described as follows:incident neutrons with the nominal wave vector ki are Bragg scattered along thenominal scattered-beam direction (kf - direction). These neutrons do not have the
2.7-6
proper energy to be Bragg-scattered by the analyser. Nevertheless some of themmay be scattered into the detector via some other process (incoherent, diffuse orinelastic scattering) taking place at the analyser position. The low efficiency of suchprocesses is offset by the high efficiency of Bragg scattering at the sample and thus,if in the course of a constant-Q or constant-ω scan a configuration such as the oneshown in Fig. 3a is encountered, a significant increase in the detector counting ratemay result. Such a process can be detected with the help of a monitor placed imme-diately before the analyser crystal, the so-called “monitor 2”.
Fig. 3(b) shows the reverse process (“type II”) where the non-Bragg scatteringoccurs at the monochromator position. This latter type of event is not detected by“monitor 2”.
Around any Bragg peak ττhkl there exist two directions in Q-space for whichspurious acoustic-like “inelastic” peaks will appear on the focused side of a con-stant-Q scan: these “dangerous directions” are those of and , where and
refer to the Bragg scattering configuration:
When searching for long-wavelength excitations, i.e. when the above picture must be extended, in such a way as to include the effect of sam-ple mosaicity.
Effects related to sample mosaicity are significant when the ratio |q|/|ττ| becomes
Q = + q qτ τwith <<
k k hkiB
fB
l− = τ
kfB
kiBkf
BkiB
2.7-7
Figure 3: Configuration for spurious inelastic peaks.
comparable to or smaller than the sample mosaic spread, expressed in radians.Qualitatively, the effects are twofold:
• the spurious acoustic-like “branches” are no longer restricted to special q-directions.
• the peaks occur symmetrically on both the focused and defocused sides of aconstant-Q scan.
Quantitatively, the spurion “dispersion relation” is found as
(4)
where the - and + signs refer to type I and type II processes, respectively.When the spectrometer is operated in the constant kf mode, the value of ki, at whichthe spurious peak should occur, is not known a priori. Since, however, eq.(1) aboveis only valid to first order in |q|/|ττ| one may replace ki by kf.
References[1] C. Stassis, In: Neutron Scattering (K. Skold, D.L. Price, eds.), Methods ofExperimental Physics Vol. 23A, Academic Press, 1986, p.369.
W.G. Stirling, K.A. McEwen, ibid Vol. 23C, p.159.
[2] L. Pintschovius In: Proceeding of the Workshop on Focusing Bragg Optics,Braunschweig, Germany, 1993. Nucl. Instr. And Meth. In Phys. Res. A338 (1994) 136.
W. Buehrer, ibid p. 44.
[3] J.Kulda & J.Saroun, Nucl. Inst. Meth. A379 (1996) 155-166.
[4] M.J. Cooper, B. Nathans, Acta Cryst. 23 (1967) 357-367.
[5] B. Dorner, Acta Cryst. A28 (1972) 319-327.
[6] M.Popovici, A.D. Stoica, M. Ionita, J. Appl. Cryst. 20 (1987) 90.
[7] J. Saroun, J. Kulda, Physica B 234-236 (1997) 1102.
[8] K. Lefman, K. Nielsen, A. Tennant, B. Lake, Physica B 276-8 (2000) 152-3.
ντ
τsiTHz k( )( )q q≈ ± ⋅22
2
2.7-8
The basics of Neutron Spin Echo
B. Farago
IntroductionUntil 1974 inelastic neutron scattering consisted of producing by some means aneutron beam of known speed and measuring the final speed of the neutrons afterthe scattering event. The smaller the energy change was, the better the neutronspeed had to be defined. As the neutrons come form a reactor with anapproximately Maxwell distribution, an infinitely good energy resolution can beachieved only at the expense of infinitely low count rate. This introduces a practicalresolution limit around 0.1µeV on back-scattering instruments.
In 1972 F. Mezei discovered the method of Neutron Spin Echo. As we will seein the following, this method decouples the energy resolution from intensity loss.
BasicsIt can be shown [1] that an ensemble of polarized particle with a magnetic momentand 1/2 spin behaves exactly like a classical magnetic moment. Entering a regionwith a magnetic field perpendicular to its magnetic moment it will undergo Larmorprecession. In the case of neutrons:
ω = γBwhere B is the magnetic field, ω is the frequency of rotation and γ =2.916kHz/Oe isthe gyromagnetic ratio of the neutron.
When the magnetic field changes its direction relative to the neutrontrajectory, two limiting cases have to be distinguished:1) Adiabatic change, when the change of the magnetic field direction (as seen by
the neutron) is slow compared to the Larmor precession. In this case thecomponent of the beam polarization which was parallel to the magnetic fieldwill be maintained and it will follow the field direction.
2) Sudden change, is just the opposite limit. In that case the polarization will notfollow the field direction. This limit is used to realize the static (Mezei)flippers.
Now let us consider a polarized neutron beam which enters a B magnetic fieldregion of length l1. The total precession angle of the neutron will be:
ϕ1 =γνB l1 1
1
2.8-1
depending on the velocity (wavelength) of the neutrons. If v1 has a finitedistribution, after a short distance (a few turns of the polarization) the beam willappear to be completely depolarized. If now the beam will go through an otherregion with opposite field B2 and length l2 the total precession angle is:
In the case of elastic scattering on the sample v1=v2= v and if B1l1 = B2l2 then ϕtot
will be zero independently of v and we recover the original beam polarization. Letus suppose now that a neutron is scattered with a small ω energy exchange betweenthe B1 and B2 region. In that case in leading order in ω :
where m is the mass of the neutron
If we put an analyzer after the second precession field and the angle between thepolarization of a neutron and the analyzer direction is ϕ, then the probability that aneutron is transmitted is cosϕ. We have to take the expectation value of overall the scattered neutrons. At a given q the probability of the scattering with ωenergy exchange is by definition S(q,ω). Consequently the beam polarizationmeasured is:
So NSE directly measures the intermediate scattering function where:
It is important to notice that t∝ λ3 so the resolution in t increases very rapidly with λ.
ImplementationIn practice reversing the magnetic field is difficult as in the middle this wouldcreate a zero field point where the beam gets easily depolarized. Instead, acontinuous horizontal field is used (conveniently produced by solenoids) and a [/2flipper starts the precession by flipping the horizontal polarization perpendicular tothe magnetic field. The field reversal is replaced by a [ flipper which reverses theprecession plane around an axis and at the end a second [/2 flipper stops theprecession and turns the recovered polarization in the direction of the analyzer.
1 3
ν
tBl
m= hγ
ν 3
cos
cos( ) ( , )
( , )( , )ϕ
γν
ω ω ω
ω ω=
∫
∫=
h Bl
mS q d
S q dS q t
3
cosϕ
ϕ γ
νωtot
Bl
m= h
3
ϕ γν
γνtot = −B l B l1 1
1
2 2
2
2.8-2
ElementsMonochromatisation:The biggest advantage of NSE is that it decouples monochromatisation from energyresolution. Whenever q resolution is not so important (or it is determined by theangular resolution rather than the wavelength distribution ), a neutron velocity selectorwith high transmission (e.g. ∆λ/λ=15% FWHM ) is the best choice. If necessary thewavelength distribution can be reduced by installing after the analyzer a crystalmonochromator (graphite or mica) and the detector in Bragg geometry.
A further option is to use time of flight (TOF) as implemented on IN15. Thefully polychromatic beam (Maxwell distribution from the reactor) can be pulsed byusing rotating disk choppers relatively far (e.g. 20m) from the detector. By the timethe neutrons reach the detector, the pulse will be completely spread and from theflight time we can calculate back that at any moment which wavelength at whatmonochromatization is detected. Collecting data in different time channels, a wholerange of wavelengths can be used.
Polarizer:This is a very essential part of the spectrometer, and generally supermirrors areused. There are two possible choices, reflection or transmission mode. As thepolariser-sample distance is in the order of 2-3m the main danger in both cases isthe unwanted increase of the incoming beam divergence from the polariser. Due tothat relatively big distance here one can loose integer factors in the neutron flux on
2.8-3
Figure 1: Sketch of a NSE spectrometer
the sample. Inherently the reflecting supermirrors are more susceptible to increasethe beam divergence, and also have the inconvenience that the reflection anglemight have to be adjusted to different wavelength if the useful range is largeenough. On the other hand the supermirrors of this type were easier to obtain.Recent progress allows the production of supermirrors on Si wafers withsufficiently good quality to make cavity like transmission devices.
As this is the most important part of the incoming beam it needs a closerexamination. If we look at the trajectory A which comes parallel with the beamaxis, we can see that it becomes very divergent after the polariser. Indeed if the
wavelength is sufficiently large, the first reflection on the supermirror is below thecritical angle and the polarization happens only when the neutron is bounced backfrom the guide, but now with a larger incoming angle. However there is trajectoryB arriving with the same large divergence and, as shown on Fig 2, will go out of thepolariser parallel with the axis. If this very divergent neutron is missing, due tomisalignement or cuts in the guide, trajectory A will not be replaced and effectivelythe center of the beam will be emptied.
and /2 flipper:The flipper action is based on the nonadiabatic limit. Using a flat coil (≈5mm thick)perpendicular to the neutron beam with a relatively strong field inside andarranging such that the horizontal field is small, the polarized neutrons goingthrough the (thin) winding do not have time to follow adiabatically the change inthe field direction. All of a sudden the beam polarization is no more parallel to themagnetic field and it will start to precess around the field. With the appropriatechoice of the field strength and direction, the neutrons will precess exactly [ or [/2by the time they fly through the flipper.In the case of TOF, the flipper current has to be modulated in time to achieve thedesired action for the wavelength which is just passing through.
2.8-4
Figure 2: “Cavity” type transmission polariser
Main precession coilsWe have seen that the Fourier time is proportional to the field integral. If we want a goodresolution, we need high field and long integration path. Both will have a drawback.Even the most symmetric solution for the main precession coil, a solenoid, will producean inhomogenous field for finite beam size. The underlying physical reason is that in themiddle of the precession coil we have a strong field and at the flippers, as was explainedabove, we need a small horizontal component. Since Maxwell we know that div(B) iszero, consequently the transition region will introduce inhomogeneities.
Making long precession coils is not necessarily optimum because thatincreases the distance between the source (polariser exit) and the sample as well asthe sample-detector distance. The counting rate will decrease with R-2*R-2.
Special attention has to be payed that at least the winding is as regular aspossible to be able to calculate the necessary corrections. This is not always anevident task as for example on IN11, the coils consist of two layers of water cooledhollow conductor, with a total number of turns 400 and max current of 600 Amps.
“Fresnel coils”How to correct the unavoidable inhomogeneities due to the finite size beam? Let usconsider the field integral difference between the trajectory on the symmetry axisand one parallel at a distance r through the solenoid. In leading order this gives:
∆ ∫ ≅ ∫
Bdzr
B z
B z
zdz
2 2
8
1
( )
( )∂∂
2.8-5
Figure 3: ?/2 and ? “Mezei” flippers simplified scheme
π/2 Flipper180° Precession around an axis at 45°
π Flipper180° Precession around a vertical axis
It can be shown also that a current loop placed in a strong field region changes thefield integral by 4[I, if the trajectory passes inside the loop and by zero if it passesoutside. To correct the above calculated r2 dependence we just have to place properlyarranged current loops in the beam. The so called Fresnel coils just do that.[1]
They are usually made by printed circuit technology preparing two spiralswhere the radius changes as r∝ , placedback to back with a thinnest possible insulator(30 micron kapton) to minimize neutronabsorption.
ResolutionWhy field inhomogeneities are bad? In fact allwe are interested in is the field integral. Wehave to use finite beam size if we want to havefinite counting rates. With finite beam size wehave different neutron trajectories arriving onthe sample and coming to the detector. If the
field integral for different trajectories are non equal then the final precession anglewill be different for different trajectories and the detected
polarization even for an elastic scatterer will be reduced to . The echomeasured on the sample will be further reduced due to the energy exchange. Usingan elastic scatterer we can measure the influence of the inhomogeneities and divideit out. (in other words while in energy space the instrumental response has to bedeconvoluted from the measured curve to obtain the sample response, in theFourier transformed space the deconvolution becomes a simple division)Nevertheless if the field integral for different trajectories is too different, then
quickly becomes zero and no matter how precisely we measure it, divisionby zero is usually problematic!
cosϕ
cosϕ
ϕ
2.8-6
Figure 4: Fresnel coil
Figure 5: Echoresponse for anelastic scatterer(resolution) andsample.
ϕ γν
γνtot
B l B l= −1 1
1
2 2
2
To put this into context lets have a look at some realistic numbers:On IN11 we have a maximum field integral of ≈3*105 Gauss·cm. With 8Åwavelength this gives a Fourier time of 28nsec. An exponential decrease with atime constant of 28nsec correspond in energy space to a Lorentzian line width of 23neV. With the incoming neutron energy (8Å) this is a relative energy exchange ofthe order of 10-5. The 360 degree phase difference between two trajectoriescorresponds to 17 Gauss·cm difference in field integral. This means a requestedrelative precision of better than 6*10-5 for all trajectories!
So what is the theoretical limit to the achievable resolution? There is no easyanswer. In principle with sufficiently precise in-beam correction coils (like theFresnels but including higher order term corrections) any resolution should bereachable. In practice we could certainly reached already an overall correctionbetter than 10-4.
If one changes the collimation conditions, the neutrons will explore differenttrajectories thus picking up different field inhomogeneities and the spectrometerresolution will change. When the resolution is pushed to its limits the number ofimportant parameters which can give artifact increases rapidly and particularattention has to be paid to avoid them.
Measurement sequenceNow we will detail how an experiment is really done. Once the aims of theexperiments are clear one has to decide what wavelength to use. Long wavelengthgives better resolution but lower flux.
Then it has to be decided at which precession field (Fourier time) the echopoints has to be measured. Depending on the problem studied, it can be adequate totake equidistant points on a logarithmic scale.
The field integral on the two side of the [ flipper must be equal to a precisionof 10-5. This is done by using an additional winding on the first precession coil, the
2.8-7
Figure 6:Resolution correctedsample response
symmetry coil. Scanning the current in this coil will go through the exact symmetrypoint producing the typical echo group as shown in Fig. 7.
The periodicity of the damped oscillation is determined by the averagewavelength and the envelope is the Fourier transform of the wavelengthdistribution. These depend only on the monochromator (velocity selector) so do notcarry information on the sample. (With one notable exception, when the sample iscrystalline and remonochromatize the beam. In that case the echo envelopebecomes much more extended).
With the echo we measure how much of the initial polarization we recover. Forelastic scatterer on an ideal instrument we recover it fully. During the scatteringprocess the initial beam polarization can change. One reason for this can be when thesample contains hydrogen which has a high spin incoherent cross section. Spinincoherent scattering changes the initial polarization from P to -1/3P. The scatteredintensity is thus composed of the sum of coherent and incoherent scattering decreasingthe up/down ratio. The relative weight of these can, and will change as a function ofscattering angle (or wave vector q). The measurement sequence thus starts with themeasurement of spin-up and spin down for the given scattering angle and samplecondition. This is accomplished by setting a guide field to maintain beam polarizationand counting with [ flipper on and [ flipper off. This will give us two points marked asUp and Down on the figure 7. It would be inefficient to measure always the wholeecho group as we are interested only in its maximum amplitude. At most it is equal to(Up-Down)/2. On an elastic scatterer we can quickly find where the center of the echogroup is as a function of the current in the symmetry coil. As we also know theperiodicity of the sinusoidal echo group it is more efficient to measure four pointsplaced around the center with [/2 steps. This will give us:
2.8-8
Figure 7: Measurement sequence
E1 = Aver+Echo*sin(φ)E2 = Aver-Echo*cos(φ)E3 = Aver-Echo*sin(φ)E4 = Aver+Echo*cos(φ)
From the four measurements Aver,Echo and φ (the phase) can be determined, andfinally
I(q,t)/I(q,0) = 2*Echo/(Up-Down)
If we are sure that the phase is zero, then it is sufficient to measure E2 and E4.However in general to monitor any possible phase drift, (due to current instability,external perturbation, .... or accidental displacement of a coil.....) usually all thefour points are measured but giving more weight to E2 and E4.
Signal and BackgroundNo general recipe can be given, besides that both should be measured and theappropriate correction done! Nevertheless let us discuss a few typical cases and forthe sake of simplicity let us take an ideal instrument (all flippers, polariser,analyzerhaving 100% efficiency, perfect field integrals).
Coherent, (spin) incoherent scattering:As was mentioned, incoherent scattering changes the beam polarization to -1/3.Let’s consider the case when the scattered intensity is composed of I= Icoh +Iincoh. Then we will have:
Up = Icoh+1/3IincohDown = 2/3IincohEcho = Icoh*fcoh(t) -1/3Iincoh*fincoh(t)
where fcoh(t),fincoh(t) denotes the time dependence (dynamics) of eachcontribution and at t=0 both are equal to one.
Our usual data treatment will give
I(q,t)/I(q,0) = [Icoh*fcoh(t) -1/3Iincoh*fincoh(t)]/[Icoh-1/3Iincoh]
This means that from Up and Down we can determine the relative weight of eachcontribution but the separate time dependence is impossible to extract (except ifsuitable model exist and even then, it is hard to achieve the necessary statisticalaccuracy) . Fortunately in many cases it is possible to minimize the incoherentcontribution by using deuterated samples. (Note the incoherent echo signal isfurther reduced by the factor 1/3) In other cases it is possible to suppose that the
2.8-9
dynamics of fcoh(t) and fincoh(t) is the same or nearly identical and then:
I(q,t)/I(q,0) = [Icoh*fcoh(t) -1/3Iincoh*fincoh(t)]/[Icoh-1/3Iincoh] = f(t)
There can be also some scattering from the sample holder, solvent, etc. Some casescan be treated in a simplified way. For example in polymer solution, scatteringfrom deuterated solvent gives coherent scattering, but it is too inelastic to be seenby NSE, which means its relaxation time is out of the time window. In this case itwill influence Up-Down but gives no echo. It is sufficient to measure Up and Downof the solvent. Similarly sample holders usually give only some elastic scattering.Again Up,Down and a few echo points in time are sufficient to establish itscontribution.
NSE vs standard inelastic instrumentsNSE not only has very high energy resolution but also has a very wide dynamicalrange. With some tricks (double echo) it can be as high as tmax/tmin = 1000 [2].With the combination of several wavelength this can be further extended by a factor10-100. We can measure energy exchanges up to 100µeV which not only overlapswith back scattering but also with Time-Of-Flight like IN5 or IN6. When to choosewhich instrument?The fundamental difference is that NSE measures in Fourier time while TOF inomega space. Consider a sample which has a strong (95% of the intensity) elasticline, and well defined but weak (5%) excitations at finite energy. While this is wellseparated in omega space (Fig. 8), on NSE the Fourier transform will give a small(5%) cosine oscillation around 0.95. Statistically this is very unfavorable (Fig. 9).Furthermore the rough wavelength distribution (around 15% FWHM) of NSE willsmear out the oscillations (t ~λ3). For that type of experiments TOF is better suited.
On the other hand when all (or most of) the intensity is quasielastic, NSE cangive more precise information on the time dependence. The main reason is that the
2.8-10
Figure 8: Spectrumin w space measuredon a Time-Of-Flightspectrometer.
deconvolution of the instrumental resolution is replaced by a simple division whichleaves the data less influenced by numerical manipulation. Below is an example ofthe relaxation time measured on magnetic nanoparticles imbedded in an aluminiummatrix (Fig. 10). With the combination of IN11 and IN15 (the NSE instrumentoptimized for long wavelength , long Fourier times) more than four decades in timecould be covered.
Studying fully incoherent scatterers is also not very favorable for NSE.Already incoherent scattering in general gives rather weak intensity, secondly the -1/3 factor in polarization decreases the statistical accuracy. These drawbacks aresomewhat lifted by now on the multidetector spin echo instruments like IN11C andSPAN in Berlin. (see below)
Finally an interesting observation. If accidentally the incoherent contributionis about three times higher than the coherent one, then we end up with no
2.8-11
Figure 9: Samespectrum than Fig. 8measured in the timedomain by NSE.
Figure 10: Relaxation spectrum of diluted magnetic nanoparticles measuredby NSE.
polarization of the scattered beam and if the dynamics of the two is the same, theexperiment is unfeasible on NSE as the echo signal will be always zero.
VariantsInelastic NSEOne could ask the question whether the method could be used in the case of well definedexcitations? The answer is yes, but with some modification. By choosing B1l1 ≠ B2l2the same Fourier back-transformation happens but around the excitation energy ω0 [1]:
;
v1 is the velocity of the incoming neutron and v2 is after the scatteringThe problem here is that around ω=0, S(q,ω) usually has a strong peak (eitherquasielastic scattering, diffuse elastic scattering and/or incoherent scattering or justinstrumental background) The solution is to use a triple axis spectrometer geometrywhich acts as a filter, pre-cutting an interval around ω0 with a coarse resolutioncompared to NSE. There is a further requirement when ω is q dependent ( like aphonon branch). We have to make the spin precession dependent on the incomingand outgoing direction. Interested readers should look up the details in ref[1]. Herewe just give an easy to understand picture. Quasielastic NSE measures the Fouriertransform from ω to time space. This is a Fourier transform in the 1D space. Nowwe switch to the (at least) 2D q- ω space. If ω is q independent (at least in the
∫ −
∫ −=
cos( ( )) ( , )
( ,( ))( , )
hγν
ω ω ω ω
ω ω ω
B lm
S q d
S q dS q t
1 1
1
3 0
0
ω ν0 12 1 1
2 2
3
2
21= −
m
h
B l
B l
2.8-12
Figure 11: Principles for measuring linewidth on a dispersion curve by NSE.
vicinity we are measuring, e.g. close to the Brillouin zone boundary) withasymmetric field integral we make the Fourier transform around a finite ω0 and weobtain the line width. Now if we do the same e.g. on an acoustical branch we willFourier transform along a vertical line. To measure the real physical line width, weneed to do the Fourier-transformation along a line perpendicular to the dispersionrelation (Fig. 11). This refocalisation of the echo works if the spin precessionbecomes dependent on the neutron direction.
Magnetic scatteringThe application of NSE to magnetic scattering has special features. One is thatferromagnetic samples can hardly be studied. The random orientation of magneticdomains introduces Larmor precessions around random axis and unknownprecession angles normally resulting in depolarisation of the beam and the loss ofthe echo signal. In some cases the application of a strong external field to align allthe domains in the same direction can help for one component of the polarization tosurvive [1]
The second case is paramagnetic scattering. If the scattering vector ( q) is inthe y direction, then in the sample only the spin components which are in the xzplane (perpendicular to q) will contribute to the magnetic scattering. As theprecession plane of the neutrons in the usual NSE geometry is the xy, the neutronswhich arrive spin parallel to q will undergo spin flip scattering, while those arrivingperpendicular to q will have 50% probability of spin flip and 50% probability ofnon spin flip scattering. Accordingly the scattered beam polarization can bedecomposed as:
We can realize that the second term is equivalent to a [ flipped beam aroundthe y axis. Consequently without a [ flipper the second term will give an echo with50% amplitude of the magnetic scattering, while using a [ flipper the first termresults in a negative echo of 50% amplitude plus a full echo from the eventualnuclear scattering as well. This means that without [ flipper only the magneticscattering gives an echo signal, which eliminates the need of time consumingbackground measurements to separate the nuclear contribution.Antiferromagnetic samples do not depolarize the beam, so in most cases they areidentical to paramagnetic samples. However if we measure a single-crystal it mighthappen to be a single domain and depending on the spin direction one can have fullecho without [ flipper or we might need a [ flipper to see any echo.
12
0
12
0
−−
+ −
P
P
P
Px
y
x
y
2.8-13
Resonance or Zero Field Neutron Spin Echo (ZFNSE)This method was introduced by Gähler & Golub [3]. We will just give a quickexplanation how to understand the basic principle. The method has clearadvantages in the case of inelastic NSE. Whether the quasielastic NSE will or willnot have better performance we will see when sufficient experience will beaccumulated with ZFNSE.
The magnetic field profile along the axis of a classical NSE schematically isthe following: Nearly zero horizontal field at the first [/2 flipper, strong (B0) fieldin the first solenoid, nearly zero at the [ flipper, again strong field in the secondprecession region, finally nearly zero again at the second [/2 flipper. In B0 theneutron precesses with the Larmor frequency ω0=γB0. Let us choose now a rotatingframe reference which rotates with ω0. In this frame the neutron seems not toprecess in the solenoids which means it sees B’0=0. However now in the originallylow field region it will rotate with ω0 which indicates the presence of a strong B0
field. The small field of our static flipper will appear to rotate also with ω0.Let us realize this configuration now in the lab frame. Inside the [ and [/2
flippers we must have a strong static field B0 and perpendicular to that a small fieldwhich rotates with ω0=γB0. We need a well localized strong magnetic fieldperpendicular to the neutron beam, and inside a smaller coil perpendicular both tothe neutron beam and to the strong field. This small coil has to be driven by aradiofrequency generator which will produce the necessary (resonance) rotatingfield. In between these flippers a strictly zero field region has to be insured to avoid“spurious” spin precession. The first prototype spectrometers all use mu-metalshielding + earth field compensation.
Multidetector variantsAt the time of writing, four instruments are developed to extend the “classical”NSE to use multidetectors, and thus reducing the data acquisition time. In all casesthe problem is that useful information can be collected in all detectors only if theecho condition can be fulfilled simultaneously for all of them. This is not astraightforward exercise as the field integrals must be equal with a precision of10-5! This strict condition can be somewhat relaxed, if we can have enoughinformation in each individual detector to make the phase correction on a one-by-one basis. This is not always possible if the scattering is weak. In that case, on anelastic scatterer, the relative phase differences between detectors can be measuredand imposed during the data treatment.The most straightforward implementation of such a multidetector is on theinstrument IN15 at the ILL and a somewhat later construction in KFA Jülich. Here
2.8-14
simply the solenoid diameters were increased to allow the use of a small 32cm x32cm multidetector. The difficulty is to make a sufficiently big Fresnel coil with thenecessary precision, without absorbing a substantial fraction of the neutron beam.On IN11 an option has been constructed, to replace the secondary solenoid with aspecially shaped sector magnet (Fig. 12). The expected field homogeneity isinherently lower, but in many cases in the high q region the best possible resolutionis not needed anyway. The gain factor in total countrate is however 23 time higher.This gain is sufficient to measure even the very unfavorable incoherent scatterers.
In Berlin a totally new approach was initiated. If one takes a Helmholtz likecoil, but the two half generates a field opposite to each other, then in the equatorialplane the field is guaranteed to be rotationally symmetric. The neutron beam comesin along a radius and the detectors are placed all around. The practical problem hereis a zero field point at the sample position which would depolarize the beam. Thesolution proposed is to move this zero point above the sample with the help of someadditional coils.
2.8-15
Figure 12: The IN11Cmultidetector option.
Useful numbersSome numbers from the ILL spin-echo instruments are given in the table.
Table 1:
The miminum time can be shorter by a factor 3-5 if double echo [2] configuration isused. The maximum reachable Fourier times can be shorter as much as a factor twowhen the detector is positioned at high scattering angles.
References[1] Mezei,F. (editor) Neutron Spin Echo Lecture Notes in Physics 128 Springer-Verlag 1979
[2] B. Farago Annual Report ILL 1988 p: 103
[3] R.Gähler, R.Golub Z.Phys, B65 (1987) 269.
AknowledgementThe author had the chance to learn most of his knowledge from F. Mezei.
2.8-16
Neutron Diffraction on Reactors
A.W. Hewat and G.J. McIntyre
1. IntroductionThe efficiency of a neutron diffractometer, whether on a reactor or a spallationsource, depends essentially on the time-averaged flux on the sample and the solidangle of the detector. Large new detectors and focusing neutron optics areincreasingly being used to take advantage of the high flux on the sample that isobtained with a continuous neutron source. In this article, we consider the design ofsingle-crystal and powder/liquids diffractometers, especially those using white beamsand large area detectors, for different types of experiment. We give the essentialformulae needed to extract structure amplitudes from the observed intensities.
2. The design of modern reactor-based diffractometersThe Shelter Island Workshop1 proposed that the intensity of a neutron diffractometer,whether on a reactor or a pulsed neutron source, depended only on the time-averagedflux on the sample, the sample volume, and the solid angle of the detector:
I ~ flux*sample*detector
2.1. The time-averaged flux on the sampleOn a reactor, the time-averaged flux on the sample can be greatly increased byusing a wide band of wavelengths, without reducing the instrument resolution for aparticular angle – the focussing angle – where the diffracted Bragg angle equals themonochromator Bragg angle (fig.1). For example, even on a high-resolutionpowder diffractometer such as D2B at ILL2, the relative spread in wavelength ∆λ/λcan be 1% or more for a ∆d/d resolution of 0.1%.
An extreme case of the use of a wide band of wavelengths is the “quasi-Laue”diffractometer such as LADI3 and VIVALDI4 at the ILL. A very large position-sensitivedetector (PSD), here a set of neutron-sensitive image plates, collects an almost “white”beam of neutrons falling on the sample, with the different wavelengths sorted out, notby time-of-flight (TOF) or by monochromator/analyser crystals, but by the diffractionconditions of the single-crystal sample itself (fig.3).
A continuous neutron intensity with a wide band of wavelengths, and the resultinghigh time-averaged flux on the sample, is the principal advantage of reactordiffractometers. On a TOF diffractometer, the flux on the sample is also rather constant
2.9-1
in time, since the sharp initial pulse is spread out at the sample such that the slowestneutrons arrive just before the fastest from the next pulse. But the time-averaged fluxmay be an order of magnitude lower; relatively long flight paths and a ∆λ/λ of 0.1% areneeded to obtain a resolution ∆d/d of 0.1%. Fortunately, the solid angle of the TOFdetector for high scattering angles (backscattering) can be large while still allowing highresolution, since ∆d/d ~∆θ.cotθ and cot θ is small for large θ. This can compensate forthe intrinsically lower flux on the sample of the TOF diffractometer, supposing that thesample volumes are similar.
2.2 The sample volume and resolutionThe sample volume limits the resolution that can be obtained with a PSD, and must there-fore be considered in the design of a diffractometer. For example, the D20 instrument atILL uses a large PSD with an angular resolution of 0.1º at a distance of 1470 mm from thesample. The optimal 0.1º resolution would then require a sample of less than 2.57 mmdiameter! Larger samples are normally measured on D20, with less than optimal resolu-tion. Higher resolution could of course be obtained by increasing further the sample-detec-tor distance, but then a very tall detector would be needed to conserve the solid angle.
2.9-2
Figure 1. The focusing of a wide band of wavelengths on the D2B high-resolution diffractometer. A large detector collects many simultaneousreflections from the polycrystalline sample.
Alternatively, high resolution can be obtained with Söller collimators as fineas 5’ as on D2B, and then much larger samples can be used, with sample volume
2.9-3
Figure 2. The first diffraction pattern (1984) from the D2B high-resolution powderdiffractometer2. The peak intensity is rather constant over the entire d-spacingrange, unlike with X-rays where scattering falls off with angle, or with time-of-flighttechniques where intensity falls for shorter d-spacings or wavelengths.
Figure 3. The dispersion of the “white” band of wavelengths on the quasi-Lauediffractometer VIVALDI. The large PSD collects many simultaneous reflectionsfrom the single-crystal sample, where each reflection corresponds to a differentwavelength.
and neutron intensity increasing with the square of the diameter. Söller collimatorsalso limit the detector solid angle, since even with 128x5’ collimators, only 6.7% ofthe 158.5º scattering angle is covered, but a sample as large as 16 mm in diametercan be used, together with tall linear-wire detectors to increase the solid angle.
Until very large, high-resolution PSD’s can be constructed, the D2B-type instru-ment is then more suitable for very high resolution, while the D20 instrument is moresuitable for very high intensity. There is of course a degree of overlap in application,depending on the amount of sample actually available, and the particular problem.
The resolution is worse at angles far from the focussing angle, but by choosing alarge focussing angle the instrument resolution can be made to match the spacingbetween Bragg peaks (fig. 4). There are few peaks at very low or very high angles, andif necessary the neutron wavelength can be increased to ~6Å to move reflections withlarge d-spacings to higher angles for better resolution. This situation is different in thecase of the backscattering TOF diffractometer, where the resolution is equally good forall d-spacings, but modern TOF instruments such as GEM at ISIS also use lower-angledetectors to increase efficiency and collect data with a variety of resolutions.
2.9-4
Figure 4. The resolution or Full Width at Half Maximum of peaks is best at thefocussing angle (140o), which is high so as to match the FWHM required toresolve adjacent peaks for example, a 16Å cubic cell2.
2.3 The detector solid angleWhen only small samples are available, as in pressure cells, or when mediumresolution is sufficient, the PSD on a reactor can be as large or larger than that on apulsed neutron source. Figure 5 shows the new 2D PSD being constructed at ILL forD19. This detector covers 30º vertically by 120º horizontally, covering a solid angle of4.sin(15o)x120/360=1.1 steradians, a record for electronic detectors.
The D19 detector is being built for measurements on single crystals and fibres,but would also be good for fast powder or liquid diffraction, with an angularresolution of 11’ (2.5 mm wide elements at 760 mm). Using a 160o detector, aDiffractometer for Rapid Acquisition over Ultra-Large Angles (DRACULA) mightbe constructed to compete with the fastest diffractometers on the future SNS pulsedsource. (See table below.)
ILL-D20 ILL-DRACULA US-SNSTime averaged sample flux 5x107 ~108 ~2.5x107
Sample volume 1 1 1Detector solid angle 0.27 sr 1.45 sr 3.0 sr
Table 1. A comparison of possible high-intensity powder diffractometers onthe ILL reactor and US spallation source, showing that the high time-averagedflux at the sample, together with a D19-type PSD detector, should allow thereactor machine DRACULA to compete with the best pulsed source.
2.9-5
Figure 5. The new 30º x 120º 2D multi-wire PSD for D19 at ILL Grenoble. Suchlarge detectors match the solid angle of the best pulsed-neutron sourcediffractometers, while also benefiting from the very high time-averaged neutronflux available on the sample at a reactor source.
Reactor sources look even more attractive when continuous white beams ofneutrons can be used, as with the quasi-Laue technique on image-plate detectors such asthose of LADI and VIVALDI at ILL (fig.6). Then the time-averaged flux on the sampleis very much greater than can be obtained either with monochromatic or pulsed neutrontechniques. As well, the image-plate detector covers a large solid angle (7.9 steradians);image plates can be curved over a cylinder around the sample, while still being read outelectronically. These diffractometers are at present very new, and could be even furtherimproved by using super-mirror optics to increase the flux on the sample.
3 The resolution function in diffractometryThe 3-D resolution function of a diffractometer depends on many factors,principally the incident-beam or incident-guide divergence, the monochromatormosaic, the sample size and mosaic, and the scattering geometry, and is
2.9-6
Figure 6. The first neutron image-plate diffractometer LADI, developed atEMBL-ILL Grenoble, showing the image plate (1) on a rotating drum (2)surrounding the crystal on its supporting rod (3). The neutron beam entersand exits through the small holes in the middle of the drum, and the remainingcomponents allow the laser and photo-multiplier tube (7-9) to scan the surfaceof the image plate to read off the diffraction pattern.
conveniently measured or calculated as an ellipsoid in terms of differentialincrements, ∆γ, ∆ω and ∆ν, in the observational variables, γ, ω and ν. Following(for the most part) the definitions of Busing & Levy5, we describe the crystalorientation by the Euler angles ω, χ and φ. To allow for out-of-plane reflections wedescribe the direction of a particular diffracted ray by γ and ν, the in-plane and out-of-plane angles, respectively, on the surface of the Ewald sphere. 2θ remains theangle between the incident and diffracted beams, so that,
cos2θ = cosγcosν. (1)
In single-crystal diffraction the resolution in ω is usually considerably betterthan that in γ, which in turn is considerably better than that in ν.
The derivation of the full 3-D resolution function is straightforward but lengthy.Since the landmark papers by Nathans and Cooper6, there have been many derivationsfor various instrumental geometries. These usually derive the function in terms ofGaussians, which facilitates the numerous convolutions particularly in three-axisgeometry with its multitude of collimators, but can disguise the effect of individualparameters. For diffraction, Schoenborn7 derived explicitly the direction in the multi-dimensional function of the contribution from each of the various instrumentalparameters. The different contributions generally act along different directions,knowledge of which may indicate how to optimise a particular measurement.
∆γ, ∆ω and ∆ν can be converted to reciprocal-space coordinates over the range ofinterest for one reflection by a homogeneous affine transformation; for powderdiffraction ∆γ (∆2θ) is then related to ∆d*. Going to shorter wavelength (λ) decreasesthe resolution in the reciprocal directions corresponding to ∆γ and ∆ν. The increasinguse of PSD’s with their capability to resolve single-crystal reflections in 3-D means thatwe can tolerate a shorter wavelength if this is needed to give better real-space resolution(dmin or Qmax). The principal rule, though, is to use as long a wavelength as possible toprofit from the λ4 dependence of the product of the reflectivities of the monochromatorand the sample. Fig. 7 shows some resolution characteristics of the ILL single-crystaldiffractometers for chemistry and physics.
3.1 ‘Conventional’ crystallographyThe aim of the experiment is to measure the integrated intensities of manyreflections, even for incommensurate and magnetic structures. We are generallyinterested in nearly equal resolution in all directions of reciprocal space, so thechoice of diffractometer is determined by which of ∆2θ, ∆ω and ∆ν is largest,which is usually ∆ν. A corollary is that instruments intended for ‘conventional’crystallography usually avoid the use of vertically focussing monochromators.
2.9-7
3.2 Phase transitions and critical scatteringHere the aim is often just to scan repeatedly a few reflections or a limited region ofreciprocal space while varying the temperature, magnetic field or pressure.Concentration of the time-averaged flux into a small wavelength band clearly gives theadvantage to the monochromatic single-crystal diffractometer. The adaptability ofmonochromatic diffractometers means that we can usually orient the sample to put thereciprocal-space direction of interest most nearly along ∆ω, and we can even accept alarge ∆ν if the flux on the sample is increased by a vertically focussingmonochromator. One important consequence of the affine transformation between ∆γ,∆ω and ∆ν, and orthogonal reciprocal-space coordinates is that there is one orientation(up or down) of a given reciprocal-lattice plane that gives best resolution along a givenline in that plane 8. The orientation is usually that which gives the larger |∆ω + ∆φcosχ| between q steps along the line of interest.
2.9-8
Figure 7. Different ILL single-crystal diffractometers optimised for differentscience: full-width at half-height (∆∆ωω) versus Q (= 4sinθθ/λλ) for typical samples andchoice of monochromators, and the approximate relative intensities. More intensityis available from other optional monochromators, at the expense of resolution.
3.3 Laue diffractionThe spatial resolution of the reflections is given by the projection of the crystalshape onto the image plate, convoluted with the divergence of the incident beam,the mosaic spread of the crystal (negligible for a crystal of good quality), and thepoint spread function of the image-plate and scanner. The divergence of theincident beam in a curved thermal guide is typically 3.5λ mrad Å full width at halfheight, which gives ~0.25° at λ = 1.5 Å, a typical wavelength of maximum flux ona thermal guide. In terms of the angle between reciprocal-lattice directions Q1 andQ2 the spatial resolution becomes,
∠Q1Q2 = w/(2R) * α * η * p (2)
where w is the cross-section of the crystal perpendicular to the diffracted beam, R isthe sample-to-detector distance, α is the beam divergence, η is the mosaic spread,and p is the point spread function. (The factor 1/2 is due to the relation θ:2θbetween the angle between scattering vectors and the angle between thecorresponding diffracted beams.) The spatial point-spread function of an image-plate detector is usually negligibly small (~150 µm). Thus for a detector of radius160 mm and a typical crystal dimension of 2 mm, the spot diameter is about 0.6° inreciprocal space. The resolution is dominated by the crystal size, and for a sphericalsample is constant over the whole Laue diffraction pattern, and over all orientationsof the crystal. Allowing for twice the spot diameter between reflection centres (toallow confident estimation of the background) we can resolve reflections up toorder 25 for a primitive cubic unit cell. A reflection of order 25 at 2θ = 180° at awavelength of 1.5 Å corresponds to a unit-cell edge of 20 Å.
4 Structure amplitudes from integrated intensitiesThe reduction of the observed integrated intensities to structure amplitudes (Fhkl
2) isthe same as for X-ray diffraction, except that no polarisation factor is required forunpolarised neutron beams. For the three diffraction methods that we discuss here, thetotal integrated intensity of one reflection is related to the structure amplitude by 9:
a) powder sample in a monochromatic beam
Ihkl = m Io(λo) ∆λo V N2 |Fhkl|2 λ o
3 [1/(4sinθ)] A* T E n s-1 (3)
b) rotating single crystal in a monochromatic beam
Ihkl = Io(λo) ∆λo V N2 |Fhkl|2 λ o
3 (1/ϖ) L A* T E n s-1 (4)
c) fixed single crystal in a polychromatic beam (Laue technique)
Ihkl = Io(λ) V N2 |Fhkl|2 λ4/2sin2θ A* T E n s-1 (5)
2.9-9
where m is the multiplicity factor for the powder reflection, λ the wavelength, V thesample volume, N the number of unit cells per unit volume, ϖ the angular scanningvelocity, L the Lorentz factor which is related to the time-of-reflection opportunity,A* the absorption correction, T the correction for inelastic phonon or thermal-diffuse scattering, and E the correction for extinction and multiple diffraction. Toobtain |Fhkl| involves measurement of Ihkl as accurately and as efficiently aspossible, followed by application of the various correction factors. For mostpowder experiments, the corrections for absorption, thermal-diffuse scattering,extinction and multiple scattering can be ignored.
The principal disadvantage of the steady-state Laue technique is that itintegrates over all wavelengths, which gives a particularly high background if thesample contains elements that give significant incoherent scattering. Theincoherently scattered neutron background is,
________ Iinc = Io (λ)Σσinc ∆λ (Vs/Vc) (1/4π) n s-1 sterad-1, (6)
where ∆λ is the wavelength band-pass and Σσinc is the total incoherent cross-section of the atoms in one unit cell, arising principally from hydrogen, which oftenconstitutes 50% of organic samples and inorganic hydrates. Here, Io (λ)Σσinc isaveraged over the band-pass. The different units for Ihkl and Iinc emphasize theadvantage of small guide divergence to reduce the point-by-point signal-to-noiseratio of the Laue diffraction pattern.
4.1 The generalised Lorentz factorThe expressions for Ihkl differ principally in the Lorentz factors. The Lorentzcorrections for monochromatic powder diffraction and for single-crystal Lauediffraction are relatively simple since the sample does not rotate. Buras & Gerward 9
show how the Lorentz factors for the three techniques are related when themonochromatic single-crystal measurement is made by an ω rotation in the equatorialplane. With a large PSD, reflections may also be observed out of the equatorial plane,for which the Lorentz factor for an ω rotation becomes:
L = 1/(sinγcosν). (7)
For some crystal structures, particularly those that are incommensurate in theirnuclear or magnetic structures, it might be preferable to perform linear scans inreciprocal space rather than pure rotational scans. Here linear refers to thetrajectory of the detector aperture through reciprocal space; for a particularreflection observed in such a scan the motion of the corresponding reciprocal-lattice point through the Ewald sphere is still along an angular trajectory. The
2.9-10
Lorentz factor is conventionally the inverse of the wavelength-independent part ofthe term dz/ds that relates the scan variable s to the normal z to the Ewald sphere inthe integral of the observed counts over s:
|Fhkl|2 α ∫ Ii(z) dz = ∫ Ii(s) (dz/ds) ds = Σi (dz/ds) ∆si Ii(si). (8)
Since the interval in s over which Ii is non-zero is usually small, (dz/ds) is usuallyevaluated at the peak maximum and taken outside the integral. However for linearscans, rather than evaluate dz/ds and then perform the integration over s, it iscomputationally more convenient to evaluate the (ds/dz)∆si directly in terms of theincremental angular shifts, ∆ωi, ∆χi and ∆φi, that effect the steps ∆si. If thereflection diffracts at the setting ω, χ, and φ 10,
λ(ds/dz)∆si = -[∆ωisinγcosν + ∆χisinωsinν+ ∆φi(cosχsinγcosν − cosωsinχsinν)]. (9)
Although this expression does not in general permit a separation of (ds/dz)∆siinto an incremental scan step and a Lorentz factor, it does correct quite generally forthe different velocities of different reciprocal-lattice points through the Ewaldsphere whatever the scan mode. |Fhkl|2 for reflections observed in linear reciprocal-space scans are thus brought to the same scale as |Fhkl|2 for reflections observed byrotational scans.
4.2 Absorption correctionWhen a monochromatic beam of neutrons with incident intensity Io passes througha homogenous plate of uniform thickness t the intensity is reduced to,
I = Io exp(-µt), (10)
which defines µ, the total linear absorption coefficient. To a reasonableapproximation the mass absorption coefficient µ/ρ, where ρ is the density of theabsorber, is independent of the physical state of the absorber, and µ/ρ for acompound is additive with respect to the µ/ρ of its elements. This gives thepractical working expression 11:
µ = (n/Vc) Σiσi (11)
where n is the number of molecules in the unit cell, Vc is the unit-cell volume, σi is theatomic absorption coefficient of atom i, defined as (µ/ρ)i (A/N) where A is the atomicweight and N is Avogadro’s number, and the summation is over all atoms in one molecule.
For neutrons in the wavelength range of interest, σ is the sum of two terms: i)true absorption, which is due to nuclear capture processes; and ii) apparent
2.9-11
absorption due to scattering, both coherent and incoherent. For single crystals theabsorption due to coherent scattering is usually considered separately as extinctionand multiple diffraction. Nuclear capture absorption varies as λ, while theincoherent apparent absorption is independent of wavelength for most elements. Anotable exception is hydrogen, for which the dependence of σinc on wavelength isroughly linear 12,
σinc = 19.2(5)λ + 20.6(9), (12)
and is usually the dominant contribution to µ for hydrogenous materials. Sears 13 liststhe values of the true absorption (σabs) and incoherent absorption (σinc) cross-sectionsrequired to calculate σ (= σabs λ/1.798 + σinc) for neutron experiments (reproducedand updated in the first article in this booklet). For measurements near nuclearresonances, it is prudent to determine the absorption coefficient experimentally.
For diffraction from a crystal bathed entirely in the beam, eq. (10) applies toeach infinitesimal volume element with t being the total path length of the incidentand diffracted beams within the crystal for that volume element. The absorption forthe whole crystal is then obtained by integration over all volume elements
I = Io (1/V)∫V exp(-µt)dV = Io A (13)
to give the transmission coefficient A, which is the inverse of the absorptioncorrection A*. Analytical integration is feasible for crystals of very regular shapesuch as spheres and cylinders 11. For faceted crystals numerical integration is theusual technique 14. These equations and methods also apply to the Laue technique,with the wavelength different for each reflection, and to very well compactedpowder samples.
There is further absorption along the incident and diffracted beam pathsoutside the crystal, the absorption in each substance, be it air, sample-capillarywalls, cryostat walls etc. also given by eq. (10). If the path length through any ofthese varies significantly with crystal orientation or detector position a correctionfor that absorption may need to be applied as well.
4.3 ExtinctionCorrection for extinction is made as in X-ray crystallography, usually following theformulae of Becker & Coppens 15, who include the crystal form via the meanabsorption-weighted path length:
T = (1/VA)∫V t exp(-µt)dV (14)
which is calculated in a similar manner to the transmission factor. To check the
2.9-12
corrections for absorption and extinction, it is recommended to make integratedintensity measurements of one or two reflections over a wide range of azimuthal angle(rotation around the scattering vector). After correction for absorption and extinction,the observed |Fhkl|2 for each reflection should be independent of the azimuthal angle.
5 Accessible regions of reciprocal space in extreme sample environmentsThe accessible region of reciprocal space for single-crystals in sampleenvironments with limited access is found by drawing the Ewald sphere when thelimit of each accessible window is in the incident beam. For the vertical andhorizontal cryomagnets often used for diffraction experiments at the ILL, theaccessible regions are as shown in Fig. 8. For a sample environment with veryrestricted access the wavelength can often be varied to bring the reflection(s) ofinterest into an accessible zone.
References1. Jorgensen, J.D., Cox, D.E., Hewat, A.W., Yelon, W.B. (1985) Nuc.Inst.Meth.
B12, 525-561.
2. Hewat, A.W. (1986) Mat.Sci.Forum 9, 69-79.
3. Cipriani, F., Castagna, J.-C., Wilkinson, C. Oleinek, P., Lehmann, M.S (1996)
2.9-13
(a) (b)
Figure 8. Reciprocal-space constructions to find the accessible regions for: (a) avertical cryomagnet with a single 25º blind arc; and (b) one of the ILL horizontalcryomagnets which has restricted openings of 15º, 32º, 30º and 32º in the equatorialplane. The heavy outer arcs denote the blind angles of the cryomagnets, the openareas the accessible regions of reciprocal space, the shaded areas the inaccessibleregions, and the black dots represent a possible reciprocal lattice plane.
J. Neutron Res. 4, 79-85.
4. Wilkinson,C., Cowan,J.A., Myles,D.A. A., Cipriani,F., McIntyre,G.J. (2002)Neutron News 13, 37-41.
5. Busing, W.R., Levy, H.A. (1967) Acta Cryst. 22, 457-464.
6. Cooper, M.J., Nathans, R. (1968) Acta Cryst. A24, 619-624, 624-627.
7. Schoenborn, B.P. (1983) Acta Cryst. A39, 315-321.
8. McIntyre, G.J., Renault, A. (1989) Physica B 156 & 157, 880-883.
9. Buras, B. Gerward, L. (1975) Acta Cryst. A31, 372-374.
10. McIntyre, G.J., Stansfield, R.F.D. (1988) Acta Cryst. A44, 257-262.
11. International Tables for X-ray Crystallography, Volume III (1968), pp 157-200.
12. Howard, J.A.K., Johnson, O., Schultz, A.J., Stringer, A. (1987) J.Appl.Cryst.20, 120-122.
13. Sears, V.F. (1992) Neutron News 3, 26-37.
14. Coppens, P., Leiserowitz, L., Rabinovich, D. (1965) Acta Cryst. 18, 1035-1038.
15. Becker, P.J., Coppens, P. (1974) Acta Cryst. A30, 129-147.
2.9-14
The Production of Neutrons
C. J. Carlile
1. THE PRODUCTION OF NEUTRONS1.1 Natural Radioactive Sources
Neutrons can be produced by fusion processes in stars, by spallation processes ofcosmic rays in the atmosphere and by the process of spontaneous fission. Other thanthat there is no natural source of neutrons. The (αα,n) reaction however, with whichChadwick was first able to isolate and identify the neutron in 1932, is the nearest. Incertain light isotopes the ‘last’ neutron in the nucleus is weakly bound and is releasedwhen the compound nucleus formed following α-particle bombardment decays.Chadwick made use of the naturally occurring α-emitter polonium-210 which
3.1-1
TECHNIQUES
Figure 1: The discovery of the neutron in 1932 by Chadwick followedexperiments of αα particle bombardment of beryllium.
decays to lead-206 with the emission of a 5.3 MeV α-particle. The bombardment ofberyllium by α-particles leads to the production of neutrons by the followingexothermic reaction which is illustrated schematically in Figure 1
He4 + Be9 → C12 + n + 5.7 MeV
This reaction yields a weaksource of neutrons with an energyspectrum resembling that from afission source and is used nowadays inportable neutron sources of the kindcommonly used for setting up neutrondetectors. Radium or americium isnormally the α-emitter in the typicaldesign shown in Figure 2.
(γγ,n) sources can also be usedfor the same purpose. In this type ofsource, because of the greater rangeof the γ-ray, the two physical
3.1-2
Figure 2: The cross-section through a portable laboratory neutron sourcecontaining radium and beryllium. A typical source is about 1cm in diameter.
Figure 3: A portable neutron source using a γγ emitterwhich by removal of the γγ source can be switched off.
components of the source can be separated making it possible to ‘switch off’ thereaction if so required by removing the radioactive source from the beryllium. Thesource illustrated in Figure 3 uses antimony-124 as the γ-emitter in the followingendothermic reaction.
Sb124 → Te124 + β− + γ
γ + Be9 → Be8 + n - 1.66 MeV1.2 Fission
Uranium-235 which exists as 0.7% of naturally occurring uranium undergoesfission with thermal neutrons with the production of, on average, 2.5 fast neutronsand the release of ~ 180 MeV of energy per fission. In a critical assembly thefission reaction becomes self-sustaining with 1 neutron required to trigger a furtherfission, 0.5 neutrons being absorbed in other material and 1 neutron able to leavethe surface of the core and be available for use. The reaction is shownschematically in Figure 4.
nthermal + U235 → 2 fission fragments + 2.5 nfast + 180 MeV
If we take the example of a 10 MW research reactor then 107 joules/sec arereleased which is 3.3 x 1017 fissions/sec at 180 MeV/fission which gives 8.5 x 1017
neutrons/sec released in the whole reactor volume. The resultant neutron energyspectrum of a reactor source is shown schematically in Figure 5.
The spectrum is composed of three distinct regions:The first is the fast neutron region where the neutrons are produced. This is
referred to as a Lamb distribution with the peak intensity occurring between 1 and 2
3.1-3
Figure 4: A schematic representation of the fission reaction.
MeV which can be described mathematically by the expression
Φ(E)dE = Φf exp (-E sinh ) dE E>0.5MeV
The intermediate energy region is known as the slowing down or epithermalregion and is characterised by a 1/E intensity distribution. Thus
Φ(E)dE = dE 200 meV < E < 0.5 MeV
describes the spectrum in this region where the source neutrons are losing energy inthe moderation process.
At low energies the neutron spectrum tends towards thermodynamicequilibrium with the moderator as the neutrons, acting like a gas, both lose energyand gain energy in collisions with nuclei in the moderator. The resultant spectrum isdescribed by a Maxwell-Boltzmann distribution with an effective temperature Tn ~300 K, always somewhat higher than the physical moderator temperature since fullequilibrium is never achieved in a finite sized moderator.
Φ(E)dE = Φth E < 200 meV
The Maxwellian peaks in intensity at an energy of about 25 meV for a roomtemperature moderator.
A stylised representation of the basic components of a research reactor, aswimming pool reactor, is shown in Figure 6. The uranium fuel is normally contained
EkT
EkT
dEn n
exp −
Φepi
E
2E
3.1-4
Figure 5: The various components in a thermal reactor spectrum (intensitiesnot to scale).
3.1-5
Figure 6: A schematic view of a research reactor showing the essential components.
EnrichedUranium
+ H2O
2.1014 2.1013 1.10132.1014
NaturalUranium
+ D2O
6.1013 2.1012 2.10124.1012
ThermalFluxΦth
n/cm2/sec
EpithermalFluxΦepi
n/cm2/sec
GammaFlux
γ/cm2/sec
Fast FluxΦf
n/cm2/sec
Table 1: Typical values of neutron and gamma fluxes for the two mostcommon fuel/moderator combinations in a 20 MW reactor.
in a number of fuel rods (although there is only one at ILL) and the critical reaction iscontrolled by neutron absorbing control rods (normally boron loaded). The moderator,which can be H2O or D2O depending on the enrichment of the fuel, is often alsoemployed as the reactor coolant. A massive radiation shield of borated concrete andsteel surrounds the reactor which protects both experimenters and instruments.
The use of natural uranium fuel with its inherently low fissile isotope contentrequires the use of heavy water D2O as the moderator since its absorption crosssection is low compared to light water H2O. When fuel enriched in the U235 isotopeis used, then the more highly absorbing H2O moderator can be used. Typical valuesof the flux components for a 20 MW research reactor are shown in table I for bothcombinations of fuel and moderator.
3.1-6
Figure 7: (a) view of thebeam tubes and the coldand hot moderators ofthe ILL reactor. (b) Thefluxes as a function ofdistance from the centreof the core are shown inthe lower graph.
Clearly, whilst the source strength of both reactors illustrated in the abovetable is the same (1.7 1018 n/sec), the flux is considerably higher in the enricheduranium/H2O case. This arises simply because of the smaller core and consequenthigher power density of this combination of fuel and moderator. The use ofenriched uranium as the fuel presents its own special problems for reactor operationhowever from the political point of view.
The principal object in designing a neutron beam reactor is to deliver themaximum neutron flux at the required energy at the neutron scattering instrument.Accordingly the noses of beam tubes are located in the region of maximum neutronintensity which can be designed to occur just outside the reactor core.
This is achieved by “under-moderating” the core which delivers a high fastneutron distribution to the surrounding reflector, where the process of moderationraises the thermal flux to a maximum. This can also be achieved for other neutronenergies by locally inserting blocks of moderating material at higher or lowertemperatures. This is most notable in the case of liquid hydrogen or deuterium coldsources. The distribution of beam tubes and the various components of the neutronspectra around the high flux reactor at the Institut Laue Langevin in Grenoble isshown in Figure 7 as an illustration. The thermal flux peaks at ~15 cm from theedge of the reactor core.
3.1-7
Figure 8: The principle behind a laboratory source of neutrons usingaccelerated deuterons onto a tritium loaded target.
1.3 Particle Accelerator SourcesNeutrons can be generated by bombarding a target with high energy particles
produced with an accelerator. Depending on the type of accelerator the neutronswill be produced continuously or in bursts. A number of types of reaction have beenemployed, of which the following are typical:
(D,T) FusionNeutrons are produced in the fusion of deuterium and tritium in the following
exothermic reaction.
D2 + T3 → He4 + n + 17.6 MeV
The neutron is produced with a kinetic energy of 14.1 MeV. This can be achievedon a small scale in the laboratory with a modest 100 kV accelerator for deuterium atomsbombarding a tritium target. Monatomic or nascent deuterium is produced by bleedingD2 gas through a heated palladium tube. The accelerated D+ ions bombard a titanium orzirconium target loaded with tritium in the form of the hydride TiT2 or ZrT2 as shown inFigure 8. Before target heating becomes a problem, continuous neutron sources of~1011 neutrons/second can be achieved relatively simply. This type of neutron source isoften used in reactor physics simulation experiments in the laboratory, for example tomeasure neutron thermalisation in graphite.
Note that this reaction is the same as that which is being exploited in thedevelopment of thermonuclear fusion reactors such as JET. In this case the energyto initiate the binding of the deuterium and tritium atoms is derived from therequired high temperature of the plasma.
3.1-8
Figure 9: A schematic representation of an electron linear accelerator whichproduces pulsed neutron beams.
Bremsstrahlung from Electron AcceleratorsEnergetic electrons when slowed down rapidly in a heavy target emit intense γ-
radiation during the deceleration process. This is known as Bremsstrahlung or brakingradiation. The interaction of the γ-radiation with the target produces neutrons via the(γ,n) reaction, or the (γ,fission) reaction when a fissile target is used.
e- → Pb → γ → Pb → (γ,n) and (γ,fission)
The Bremsstrahlung γ energy exceeds the binding energy of the “last” neutron inthe target. This reaction is comparable to the (γ,n) reaction in Sb124 referred to above. Aneutron source such as this is achieved with a moderately large purpose-built electronlinear accelerator or “linac” which produces electrons in bursts at frequencies ofbetween 25 Hz to 250 Hz at energies ~150 MeV. Examples of target materials aretungsten, lead or depleted uranium. Such a source is illustrated in Figure 9.
A source strength of 1013 neutrons/second produced in short (i.e. < 5 µs)pulses can be readily realised. This source strength is adequate for specialisedneutron scattering measurements as for example was done at the Harwell, Torontoand Hokkaido linacs, and also at the Frascati cyclotron source LISONE.
Spallation from Energetic ProtonsThe process of spallation is a common nuclear reaction occurring for high
energy particles bombarding heavy atoms. The reaction occurs above a certainenergy threshold for the incident particle, which is typically 5 - 15 MeV. The
3.1-9
Figure 10: A schematic representation of the spallation reaction.
reaction, illustrated in Figure 10 is a sequential one involving the incorporation ofthe incident particle (say a proton) by the target nucleus followed by an internalnucleon cascade, an internuclear cascade where high energy neutrons are ejected,and an evaporation process where the target nuclei de-excite by the emission ofseveral low energy neutrons and a variety of nucleons, photons and neutrinos.
The reaction is multifarious and in general involves several different targetnuclei. It is often likened to a cannon ball careering through a wooden-hulled shipalthough it is a geological term representing the splitting of rocks. In the spallationreaction 20 to 30 neutrons per incident particle can typically be generated. The energyreleased per neutron produced is quite low being ~55 MeV again dependent upon theincident particle energy and the target nucleus, and whether or not fission processesplay a part. The neutron source strength can be significantly increased by using afissile target, normally depleted uranium. The source spectrum from an 800 MeVproton spallation reaction on both a lead target (with no fission) and a uranium-238target (where fast fission contributes to the neutron intensity) is shown in Figure 11.
The cascade processes account for only ~3% of the source neutrons but since
3.1-10
Figure 11: The spectrumof neutrons emittedfrom heavy metaltargets by the process ofspallation indicating thedifference when fertilematerial is part of thetarget.
they are produced at energies up to the energy of the incident protons these extremelypenetrating neutrons dominate the shielding requirements of the source.
In practice a spallation source is usually realised with accelerated protons.These can be produced in a number of ways, for example:
1. Linear Accelerators such as LAMPF at Los Alamos which are high current, highduty cycle accelerators resulting in either long pulses or at frequencies too high tobe utilised effectively in neutron scattering instruments. Accordingly particlestorage rings, which compress the long pulses, are needed.
2. Cyclotrons such as SINQ at PSI near Zurich which produce a continuousbeam of neutrons via the spallation reaction, and
3. Synchrotrons such as ISIS in the UK. Synchrotrons have operated with lowcurrents until relatively recent times with the implementation of rapid cyclingtechniques and weak focusing magnets. Narrow neutron pulses (< 1 µs) can beproduced, due to the single turn proton beam extraction method, at a modest dutycycle (50 Hz) which is well-suited to the neutron scattering instrumentation. Atypical schematic layout of a synchrotron-based pulsed spallation neutron sourceis shown in Figure 12. Multi-turn-injection into the synchrotron is achieved usingmore modest linear accelerators for negative H- ions. At the point of injection thebeam passes through a thin electron stripper foil such as alumina and the resultantprotons enter the synchrotron for further acceleration. A neutron source strength of~ 5 1016 fast neutrons/second can be produced in a pulsed manner by this method.
3.1-11
Figure 12: The principle of operation of a spallation neutron source using acombination of linear accelerator and synchrotron accelerator.
2. MODERN NEUTRON SOURCES2.1 The Quest for Higher Intensities
Neutron scattering techniques suffer from low data rates particularly incomparison with x-ray diffraction or infra-red spectroscopy. Accordingly mucheffort has been concentrated on improving the available intensities of neutronsources. In Figure 13, originally due to Carpenter, the effective thermal neutronflux of various neutron sources has been plotted chronologically. The precisepositions of various points on this graph and the use of average flux for continuoussources and peak flux for pulsed sources has been and still is the subject of debate,without even mentioning the multitude of questionable assumptions made inattempting to compare such complex facilities using only one variable.
Nevertheless the visible trends are genuine and quite clearly fission reactors,having improved rapidly in the five years following the success of the first criticalassembly at Chicago in 1942, have reached a slowly increasing quasi-asymptotearound 2 1015 n/cm2 sec. Technical advances in instrumentation, for exampleneutron guides, focusing monochromators and area detectors have howeverensured a steady rise in data acquisition rates. Pulsed sources on the other handhave not yet reached saturation of the source flux and much potential remains to berealised with this type of neutron source before such limits are approached. Themost advanced spallation source currently under construction is the 1MW SNS
3.1-12
Figure 13: The effective neutron flux of various neutron sources from thediscovery of the neutron in 1932 until the present day.
(spallation neutron source) at Oak Ridge in Tennessee which is scheduled togenerate its first neutrons in 2006. A design study has just been completed for a 10MW European Spallation Source which promises source fluxes 30 times those ofthe 160 kW ISIS source near Oxford. The layout of the ESS is shown in Figure 14.
An important limiting factor in determining the maximum neutron output of aparticular type of source is the rate of removal of the heat deposited in the target bythe nuclear reaction. Table II shows the energy released per useful neutron in thevarious reactions discussed earlier. Spallation releases 3 times less energy thanfission which in turn releases 10 times less energy than photoneutron reactions(Bremsstrahlung). Controlled thermonuclear fusion reactors offer a future promiseof neutrons for yet lower releases of energy, and consequently the potential forhigher intensities.
If additionally the neutron generating reaction is pulsed in nature, the heatdeposited in the target can be substantially reduced (perhaps by a factor of 20)compared with a continuous source of the same equivalent source strength.
3.1-13
Figure 14: A representation of the layout of the proposed European SpallationSource with two shared pulse targets. The current design includes a long pulsetarget fed directly from the Linac.
3.1-14
Process Example Neutron Yield Energy ReleasedMeV / neutron
1. (α,n) reaction Radium-beryllium 8 10-5 / alpha 6,600,000Laboratory source
2. (D,T) fusion 400 keV deuterons 4 10-5/deuteron 10,000on tritium-loaded
titanium
3. Electron 100 MeV electrons 5 10-2/electron 2,000Bremsstrahlung on uranium& photofission
4. Fission U235 (n, fission) in 1 / fission 180nuclear reactor
5. Spallation 800 MeV protons 30 / proton 55on uranium
6. (D,T) fusion Laser fusion in 1 / fusion 18controlled
thermonuclearreactor
Table 2: The neutron yield and heat released per useful neutron from thevarious reactions used to generate neutrons.
Figure 15: A cut-away view of the ISISpulsed neutron facility.
Provided that the peak flux of the pulsed source can be utilised for a highproportion of the time frame between pulses this substantial gain can be achievedwith no decrease in data rate at the neutron scattering instrument. Conversely forthe same rate of heat deposition in the target a pulsed source will, in general,achieve a higher data rate than the continuous source.
2.2 ISIS at the Rutherford Appleton LaboratoryA cut-away view of the pulsed neutron source ISIS at the Rutherford Appleton
Laboratory in the UK is shown in Figure 15. It comprises a series of acceleratorswhich produce an intense 800 MeV pulsed beam of protons which is incident on atantalum target, producing fast neutrons by the spallation process.
We can divide the facility into nine sections for the purpose of describing it.
1. A pre-injector high voltage generator of the Cockcroft-Walton potentialdivider type. This produces pulsed voltages at 665 kV and 50 Hz, delivering acurrent of 40 mA in 500 µs bursts to ....
2. ... an H- ion source. The ion source produces negative hydrogen ions H- in acaesium vapour discharge fed with monatomic hydrogen atoms generated bypassing hydrogen gas through a heated nickel tube. The H- ions sitting at the665 kV potential of the pre-injector are driven towards the first element of ....
3. ... a linear accelerator injector of the Alvarez type consisting of a series ofpotential gaps increasing in length along the linac as the H- ions gain energy.
3.1-15
Figure 16: The principle of injection of an H- beam into a closed orbit synchrotron.
The linac accelerates the H- ions to 70 MeV producing a current of 20 mA in500 µs bursts every 20 ms (i.e. at 50 Hz). The H- ion beam enters thesynchrotron via....
4. ... an injection straight where a 0.25 µm thick foil of alumina Al2O3 strips theelectrons from the H- ions and the resulting protons are trapped in themagnetic field of the synchrotron and join the acceleration process. The use ofH- ions and multi-turn injection allows the current in the synchrotron to beincreased up to the space-charge limit. Thus H- ions and protons from earlieron in the injection process pass through the same magnetic field at the stripperfoil but with equal and opposite curvatures, as shown in Figure16.
5. The synchrotron is 52 metres in diameter in the shape of a ten-sided polygon.Each of the ten sides has common magnet elements for focusing the protonbeam and for bending it by 36° into the next straight section.
Each straight has an individual purpose. The first straight is used for injectionas we have seen above. Six straights are used for acceleration purposes, eachcontaining a radio frequency ferrite-cored accelerating cavity which feedsenergy, appropriately phased, into the proton beam. As the beam gains energyso the frequency of the RF cavities is increased to give the proton beam aslight increase in energy on each pass. The six RF cavities accelerate the beamto 800 MeV at which stage the protons are circulating in two bunches onopposite sides of the synchrotron. Each bunch is 90 nsec wide and separatedfrom the second bunch by 210 nsec. There are 2.5 x 1013 protons per pulse inthe accelerated beam which is equivalent to an average current of 200 µA.Two straights of the synchrotron are for vacuum pumping ports - the ring isheld at a vacuum of 5 x 10-8 mbar - and the final straight is ....
6. ... the extraction straight which contains a set of fast acting kicker magnets.These magnets are powered at the end of the acceleration process and the twobunches of protons in the ring are extracted simultaneously from thesynchrotron into ....
7. ... the extracted proton beamline. The proton pulse is then guided byquadrupole magnets along an 80 metre vacuum tank to...
8. ... the target station where it strikes the tantalum target thus producing a burstof fast neutrons by the spallation process. The target is surrounded by fourmoderators – two are ambient water, one is liquid methane at 110 Κ, and oneis supercritical hydrogen vapour at 25 K - and a beryllium reflector toconcentrate the neutron flux in the region of the moderators. The moderatedneutrons pass through shutters and beam lines to reach ....
3.1-16
9. ... the neutron scattering instruments. There are 18 beam holes on ISIS, nineon either side of the target station. The present (March 2002) layout ofinstruments is shown in Figure 17.
2.3 The High Flux Reactor at the Institut Laue LangevinA plan view of the high flux reactor at the ILL in Grenoble and the associated
neutron instruments is shown in Figure 18. A short discussion of the reactor coreand the coolant system has taken place earlier in the paper.
The reactor core consists of a single fuel element made from highly enriched(93%) uranium-235 with a mass of ~ 8.57 kg. The core has a diameter of 40 cm andis controlled by a single, central control rod. A power of 58 MW is generated whichis removed by pumping water through the fins of the fuel element. At equilibriumthe temperature of the fuel element is 50 °C. The core is surrounded by a D2Oreflector vessel of diameter 2.5 m and outside the reflector is a tank of H2O and aconcrete radiation shield. There are also three purpose-built moderators at differenttemperatures to provide a wide range of neutron spectra for the variousinstruments. These are a hot moderator, which is a graphite block at 2500°Cproviding peak flux at 200 meV, and two cold moderators, one being a spherical 25
3.1-17
Figure 17: A plan view of the components of the ISIS pulsed neutron sourcewith the instruments arranged around the tantalum target. Upstream is themuon target with its instrument suite.
litre vessel containing liquid deuterium at 20 K with its peak intensity at 5 meV. Inaddition there is an ultra cold neutron source which generates neutrons atwavelengths of 1000 Angstroms.
The neutron beam tubes pass through the biological radiation shield of the reactorinto the region of highest thermal flux in the D2O reflector arrayed around the core ofthe reactor. Some of the beam tubes are radial and thus view the core itself whilstothers are tangential to the core. The former have a more intense neutron flux than thelatter but with the disadvantage of a higher gamma flux also. There are also threebunches of neutron conducting tubes, or guides, which view respectively the ambientreflector and transmit a thermal neutron spectrum, and the two cold sources andtransmit a low energy or cold neutron spectrum. These guides transmit the neutronswithout a significant loss in intensity to regions distant from the reactor (40 m to 140m) where backgrounds are low and space is available to more than treble the neutroninstrumentation which can be installed close to the reactor.
2.4 A brief Comparison of Pulsed Sources and Reactor SourcesTo hazard a comparison of different neutron sources is perhaps unwise given
the large number of committed individuals and vested interests. Nevertheless my
3.1-18
Figure 18: A schematic layout of the instruments at the ILL showing the twoguide halls.
personal list is given in table III. In the end it is the science which is done on thedifferent instruments at the various sources which is important and this isdependent upon many other issues than a simple comparison will provide.
3. Further Reading
Neutron Physics by K H Beckurts & K Wirtz, Springer Verlag, 1964.
The Elements of Nuclear Reactor Theory S Glasstone & M C Edlund, VanNostrand, 1952.
Pulsed Spallation Neutron Sources for Slow Neutrons J M Carpenter, NuclearInstruments and Methods, (1977), 145, 91-113.
Spallation Neutron Sources for Neutron Beam Research G Manning, ContemporaryPhysics, (1978), 19, 505-529.
The Neutron & the Bomb Andrew Brown 1998 Oxford University Press
3.1-19
Pulsed Sources1. More high energy neutrons2. Produce neutrons in bursts & measurewhen source is off. Lower backgrounds!3. Pulsed Operation
4. Higher Φpeak possible
5. Sharp pulses give high resolution6. Pulse shape asymmetric, resolutionfunction asymmetric7. Must use Time of Flight methodsconstrained to source frequency8. Horizons still to be explored.9. Seen as environmentally friendly
Reactors1. More low energy neutrons2. Easier to shield against fission neu-trons. Lower backgrounds!3. Continuous Operation
4. Higher Φaverage difficult
5. Resolution can be adapted to problem6. Resolution function symmetric, nor-mally gaussian7. Complete flexibility
8. Mainly tried & tested techniques9. Seen as environmentally unfriendly
Table 3: Frequently stated advantages of pulsed sources and reactors. Manyof those advantages are subjective and therefore the table makes no claim tobe consistent.
If peak ~ Φaverage then data rates are “equal”
Neutron Optics
Ian S. Anderson
1. IntroductionOwing to the low primary flux of neutrons, the beam definition devices that playthe role of defining the beam conditions (direction, divergence, energy,polarisation, etc.) have to be highly efficient. The following sections give a (non-exhaustive) review of commonly used beam definition devices. A more detailedreview may be found in [1].
2. CollimatorsA collimator is perhaps the simplest neutron optical device and is used to define thedirection and divergence of a neutron beam. The most rudimentary collimatorconsists of just two slits or pinholes cut into an absorbing material and placed oneat the beginning and one at the end of a collimating distance L. The maximumbeam divergence that is transmitted with this configuration is
αmax = (a1 + a2)/L , (1)
where a1 and a2 are the widths of the slits or pinholes.Such a device is normally used for small-angle scattering and reflectometry.
To avoid parasitic scattering by reflection from slit edges, very thin sheets of ahighly absorbing material (e.g., gd or cd foils) are used as the slit material. In caseswhere a very precise edge is required, cleaved single-crystalline absorbers such asgadolinium gallium garnet (GGG) can be employed.
As can be seen from eqn. 1, the divergence from a simple slit or pinholecollimator depends on the aperture size. To collimate (in one dimension) a beam oflarge cross section within a reasonable distance L, Soller collimators, composed ofa number of equidistant neutron-absorbing blades separated by spaces, are used.The transmission, τ, of such a collimator depends on the thickness of the blades tcompared with the width of the transmission (spaces) channels s:
Blades must be as thin and as flat as possible. If their surfaces do not reflectneutrons, the angular dependence transmission of the function is close to the idealtriangular form, and transmissions of 96% of the theoretical value can be obtained
τ =+s
s t
3.2-1
with 10´ collimation. Alternatively, thin single-crystal silicon (sapphire or quartzare also suitable) wafers, coated with an absorbing layer (e.g., gd), can also be usedto construct microcollimators. The transmission losses through silicon can beminimized by choosing very thin silicon wafers (≈200 µm) coated with a fewmicrons of gd. The principal potential gain of this design, however, is thepossibility to build collimators with reflecting walls. If the blades of the Sollercollimator are coated with a material whose critical angle of reflection is equal toαmax/2 (for one particular wavelength), then a square angular transmission functionis obtained instead of the normal triangular function, doubling the theoreticaltransmission.
Soller collimators are often used in combination with single-crystal mono-chromators to define the wavelength resolution of an instrument. The Sollergeometry is only useful for one-dimensional collimation. For small-anglescattering applications where two-dimensional collimation is required, aconverging “pepper pot” collimator can be used. [2]
Cylindrical collimators with radial blades are sometimes used to reducebackground scattering from the sample environment. This type of collimator isparticularly useful for use with position-sensitive detectors and may be oscillatedabout the cylinder axis to reduce the shadowing effect of the blades. [3]
3. Crystal MonochromatorsBragg reflection from crystals is the most widely used method for selecting a well-defined wavelength band from a white neutron beam.
To obtain reasonable reflected intensities and to match typical neutron beamdivergences, crystals that reflect over an angular range of 0.2 to 0.5 degrees aretypically employed. Traditionally, mosaic crystals are preferred over perfectcrystals, although reflection from a mosaic crystal gives rise to an increase in beamdivergence with a concomitant broadening of the selected wavelength band. Thus,collimators are often used together with mosaic monochromators to define theinitial and final divergences and therefore the wavelength spread.
Because of the beam broadening produced by mosaic crystals, elasticallydeformed perfect crystals and crystals with gradients in lattice spacings may bemore suitable candidates for focusing applications since the deformation can bemodified to optimise focusing for different experimental conditions [4]. Perfectcrystals are used commonly in high-energy-resolution backscattering instruments,interferometry, and in Bonse-Hart cameras for ultrasmall-angle scattering [5].
Mosaic crystals show a much broader diffraction profile compared withperfect crystals with a lower peak reflectivity. An ideal mosaic crystal is assumed to
3.2-2
comprise an agglomerate of independently scattering domains or mosaic blocksthat are more or less perfect but small enough that primary extinction does notcome into play. Additionally, the intensity reflected by each block can be calculatedusing the kinematic theory of Zachariasen. [6]
One assumes that the mosaic blocks are oriented almost parallel to the crystalsurface (for Bragg case) following a distribution W (θ−θB), with θ being the angleformed by the incident beam and the Bragg planes and θB being the Bragg angle.The full-width-at-half-maximum η of this distribution is called the mosaic spreador mosaicity. The multiple Bragg reflections in a mosaic crystal and the concept ofsecondary extinction are summarised by the Darwin equations. An exact and verygeneral solution of these equations has been given by Sears [7]. The physical
quantities that govern diffraction by a mosaic crystal are the absorption coefficient,
µ, and the scattering coefficient . The Q factor is given by
, where λ is the wavelength, Fhkl is the structure factor, and V0
is the unit cell volume. If we define and , with d as the
crystal thickness and ϕ as the angle formed by the incident beam and the surface,
the Sears’ equations for the reflected and transmitted beam in symmetric Laue
(transmission) and Bragg (reflection) geometries are
(2)
(3)
(4)
(5)
The ideal monochromator material should have a large scattering lengthdensity, low absorption, incoherent and inelastic crosssections, and should beavailable as large single crystals with a suitable defect concentration. Relevantparameters for some typical neutron monochromator crystals are given in Table 1.
Because of the higher reflectivities that can be obtained, neutron monochromatorsare usually designed to operate in reflection geometry rather than transmissiongeometry, for which the optimisation of crystal thickness is only achieved for a smallwavelength range.The maximum peak reflectivities and the optimum thicknesses for
Ta a b
a a b a a b a b a a bBragg symm = +
+ + + + +( )
( ) cosh ( ) ( )sinh ( )
2
2 2 2
Rb
a a b a a b a bBragg symm =
+ + + +( ) coth ( ) ( )2 2
Laue symm
bT = +( )−12
1 2-e ea
Laue symm
bR = −( )−12
1 2-e ea
b d= σ ϕ/sina d= µ ϕ/sin
Q F Vhkl B= ( )λ θ3 202 2/ sin
σ θ θ= −( )[ ]Q W B
3.2-3
3.2-4
Tab
le 1
: So
me
impo
rtan
t pr
oper
ties
of
mat
eria
ls u
sed
for
neut
ron
mon
ochr
omat
orcr
ysta
ls (
in o
rder
of i
ncre
asin
gun
it-c
ell v
olum
e).
some selected reflections are shown in Figs. 1 and 2, respectively.The reflection from a mosaic crystal in reciprocal space is visualised in Fig. 3.
An incident beam with small divergence is transformed into a broad exit beam. Therange of k vectors, ∆k, selected in this process depends on the mosaic spread, η, andthe incoming and outgoing beam divergences α1 and α2:
∆k/k = ∆τ/τ + cot(θ)α (6)
where τ is the crystal reciprocal-lattice vector (τ = 2π/d) and α is given by:
α = (7)
The resolution can therefore be defined by collimators, and the highest resolution isobtained in backscattering, where the wavevector spread depends only on theintrinsic ∆d/d of the crystal.
222
21
222
221
22
21
4η+α+α
ηα+ηα+αα
3.2-5
Figure 1: Peak reflectivity for some typical monochromator crystalscalculated using an intrinsic mosaic of 0.1°.
In some applications, the beam broadening produced by mosaic crystals canbe detrimental to the instrument performance. An interesting alternative is agradient crystal, that is,. single crystals with a smooth variation of the interplanarlattice spacing along a defined crystallographic direction. As shown in Figure 3, thediffracted phase space element has a different shape from that obtained from amosaic crystal. Gradients in d-spacing can be produced in various ways: thermalgradients [8], vibrating crystals by piezoelectric excitation [9], and mixed crystalswith concentration gradients (e.g., Cu-Ge [10] and Si-Ge. [11]).
Both vertically and horizontally focusing assemblies of mosaic crystals areemployed to make better use of the neutron flux when making measurements on smallsamples. Vertical focusing can lead to intensity gain factors between 2 and 5 withoutaffecting resolution (real-space focusing) [12]. Horizontal focusing changes the k-
3.2-6
Figure 2: Wavelength dependence of the optimum thickness for variousmonochromator reflections.
3.2-7
Figure 3: Reciprocal-lattice representa-tion of the effect of a monochromatorwith reciprocal-lattice vector t on thereciprocal-space element of a beam withdivergence a. (a) For an ideal crystal witha lattice constant width ∆ττ; (b) for amosaic crystal with mosaicity h, showingthat a beam with small divergence, a, istransformed into a broad exit beam withdivergence 2h + a; (c) for a gradientcrystal with interplanar lattice spacingchanging over ∆ττ, showing that the diver-gence is not changed in this case.
Figure 4: In a curved neutronguide, the transmissionbecomes λλ dependent: (a) thepossible types of reflection(garland and zigzag), thedirect line-of-sight length, thecritical angle θθ*, which isrelated to the characteristicwavelength λλ* = θθ*
(b) Transmission across theexit of the guide for differentwavelengths, normalized tounity at the outside edge. (c)Total transmission of the guideas a function of λλ.
πNbcoh
space volume that is selected by the monochromator through the variation in Braggangle across the monochromator surface (k-space focusing) [13]. The orientation of thediffracted k-space volume can be modified by variation of the horizontal curvature sothat the resolution of the monochromator may be optimised with respect to a particu-lar sample or experiment without loss of illumination. Monochromatic focusing can beachieved. Furthermore, asymmetrically cut crystals can be used, allowing focusingeffects in real space and k-space to be decoupled. [14]
4. Mirror Reflection DevicesThe refractive index, n, for neutrons of wavelength λ propagating in a nonmagneticmaterial of atomic density N is given by the expression
= , (8)
where bcoh is the mean coherent scattering length, and m is the linear absorptioncoefficient. Values of the scattering-length density N.bcoh for some commonmaterials are reported in Table 2. The refractive index for most materials is slightlyless than unity so that total external reflection can take place. Thus, neutrons can bereflected from a smooth surface, but the critical angle of reflection, γc, given by:
γc = (9)
is small so that reflection can take place only at grazing incidence. The criticalangle for nickel for example is 0.1° Å-1.
Because of the shallowness of the critical angle, reflective optics aretraditionally bulky, and focusing devices tend to have long focal lengths. In somecases however, depending on the beam divergence, a long mirror can be replacedby an equivalent stack of shorter mirrors.
Neutron GuidesThe principle of mirror reflection is the basis of neutron guides that are used totransmit neutron beams to instruments situated up to 100 m away from thesource [15].
A standard neutron guide is constructed from boron glass plates assembled into arectangular cross section, the dimensions of which may be up to 200 mm high by 50 mmwide. The inner, reflecting surface of the guide is coated with approximately 1200 Å ofnickel, 58Ni (γc = 0.12 Å-1), or a “supermirror” (described subsequently). The guide isusually evacuated to reduce losses from absorption and scattering of neutrons in air.
λNbcoh
π
µπλ
−
4i
πλ
2
.1
2cohbN
−βη i−∂−= 1
3.2-8
Theoretically, a neutron guide that is fully illuminated by the source willtransmit a beam with a square divergence of full width 2γc , in both the horizontaland vertical directions so that the transmitted solid angle is proportional to λ2. Inpractice, because of imperfections in the assembly of the guide system, thedivergence profile is closer to Gaussian at the end of a long guide.
Because neutrons could undergo a large number of reflections in the guide, it isimportant to achieve a high reflectivity.The specular reflectivity is determinedby the surface roughness, and, typically,values in the range 98.5 to 99% areachieved. Further transmission lossesoccur because of imperfections in thealignment of the sections that make upthe guide.
The great advantage of neutronguides, in addition to the transport of neu-trons to areas of low background, is thatthey can be multiplexed, that is, one guidecan serve many instruments. This isachieved either by deflecting only a partof the total crosssection to a given instru-ment or by selecting a small wavelengthrange from the guide spectrum. In the lat-ter case, the selection device (usually a
3.2-9
Figure 5: Illustration ofhow a variation in thebilayer period can beused to produce amonochromator, abroad-band device, or asupermirror.
Material58Ni
DiamondNickelQuartz
GermaniumSilver
AluminumSilicon
VanadiumTitanium
Manganese
Nbcoh (10-6 Å-2)
13.3111.719.43.643.643.502.082.08-0.27-1.95-2.95
Table 2: Scattering length densitiesfor some common materialstypically used in neutron optics.
crystal monochromator) must have a high transmission at other wavelengths.If the neutron guide is curved, the transmission becomes wavelength
dependent. The characteristic wavelength, λ*, at which the theoretical transmissiondrops to 67% is related to the characteristic angle θ* = in the following way:
λ∗ = θ*, (10)
where a is the guide width, and ρ is the radius of curvature. For wavelengths less than λ∗ ,neutrons can be transmitted only by “garland” reflections along the concave wall ofthe curved guide. Thus, the guide acts as a low-pass energy filter as long as its length islonger than the direct line-of-sight length L1= . The line-of-sight length can bereduced by subdividing the guide into a number of narrower channels, each of whichacts as a mini-guide. The resulting device, often referred to as a neutron bender sincedeviation of the beam is achieved more rapidly, is used in beam deviators
MultilayersSchoenborn and coworkers [16] first pointed out that multilayers, comprisingalternating thin films of different scattering-length densities (N.bcoh), act like two-dimensional crystals with a d-spacing given by the bilayer period. With moderndeposition techniques (usually sputtering), uniform films of thickness ranging from~ 20 Å to a few 100 Å can be deposited over large surface areas of the order of 1 m2.Owing to the rather large d-spacings involved, the Bragg reflection from multilayersis generally at grazing incidence so that long devices are required to cover a typicalbeam width. Alternatively, a stacked device must be used. However, with judiciouschoice of the scattering-length contrast, the surface and interface roughness, and thenumber of layers, reflectivities of close to 100% can be reached.
Fig. 5 illustrates how variation in the bilayer period can be used to produce a
8.a.ρ
cohNbπ
3.2-10
Figure 6: Measuredreflectivity of an m = 4supermirror. The reductionin reflectivity at low values ofm is an artifact caused byunderillumination of thesample during themeasurement.
2 aρ
monochromator (the minimum ∆λ/λ that can be achieved is of the order of 1%), abroad band device or a “supermirror,” so called because it is composed of aparticular sequence of bilayer thicknesses that in effect extends the region of totalmirror reflection beyond the ordinary critical angle. [17] Nowadays supermirrorscan be produced which extend the critical angle of nickel by a factor, m, between 3and 4 with reflectivities better than 90% (Fig. 6). Such high reflectivities enablesupermirror neutron guides to be constructed with flux gains compared with nickelguides close to the theoretical value of m2.
The choice of the layer pairs depends on the application. For non-polarisingsupermirrors and broad band devices, the nickel/titanium pair is commonly useddue to the high contrast in scattering density, while for narrow bandmonochromators a low contrast pair such as tungsten/silicon is more suitable.
Capillary opticsCapillary neutron optics, in which hollow glass capillaries act as waveguides arealso based on the concept of total external reflection of neutrons from a smoothsurface. The advantage of capillaries, compared with neutron guides, is that thechannel sizes are of the order of a few tens of micrometers, so that the radius ofcurvature can be significantly decreased for a given characteristic wavelength (seeeqn. 10). Thus, neutrons can be efficiently deflected through large angles resultingin a more compact optical system.
Two basic types of capillary optics exist, and the choice depends on the beamcharacteristics required. Polycapillary fibers are manufactured from hollow glasstubes several centimetres in diameter, which are heated, fused, and drawn multipletimes until bundles of thousands of micrometer-sized channels are formed havingan open area of up to 70% of the crosssection. Fiber outer diameters range from300 to 600 mm and contain hundreds or thousands of individual channels withinner diameters between 3 and 50 µm. The channel cross section is usuallyhexagonal, though square channels have been produced, and the inner channel wallsurface roughness is typically less than 10 Å rms, giving rise to very highreflectivities. The principal limitations on transmission efficiency are the open area,the acceptable divergence (note that the critical angle for glass is 1 mrad/Å), andreflection losses from absorption and scattering. A typical optical device willcomprise hundreds or thousands of fibers threaded through thin screens to producethe required shape.
The second type of capillary optic is a monolithic configuration. Theindividual capillaries in monolithic optics are tapered and fused together so that noexternal frame assembly is necessary. Unlike the multifiber devices, the inner
3.2-11
diameters of the channels that make up the monolithic optics vary along the lengthof the component, resulting in a smaller, more compact design. Capillary optics canbe used in lenses to focus or collimate a neutron beam [18] or simply as a beambender.
FiltersNeutron filters are used to remove unwanted radiation from the beam while
maintaining as high a transmission as possible for neutrons of the required energy.Two major applications can be identified: removal of fast neutrons and γ-rays fromthe primary beam and reduction of higher-order contributions (λ/n) in thesecondary beam reflected from crystal monochromators. In this section, weconsider nonpolarising filters, that is, those whose transmission and removal crosssections are independent of the neutron spin. Polarising filters are dealt with in thesection concerning polarizers.
Filter action relies on a strong variation of the neutron cross section withenergy, usually either the wavelength-dependent scattering cross section ofpolycrystals or a resonant absorption cross section. Following Freund [19], the totalcross section determining the attenuation of neutrons by a crystalline solid can bewritten as a sum of three terms
σ = σabs + σtds + σbragg (11)
Here, σabs is the true absorption cross section, which at low energy, away fromresonances, is proportional to E-1/2. The temperature-dependent thermal diffusecross section, σtds, describing the attenuation caused by inelastic processes can besplit into two parts depending on the neutron energy. At low energy, E ≤ kbΘD,where kb is Boltzman’s constant and ΘD is the Debye temperature, single phononprocesses dominate, giving rise to a cross section σsph, which is also proportional toE-1/2. The single-phonon cross section is proportional to T7/2 at low temperaturesand to T at higher temperature T.
At higher energies E ≥ kbΘD, multiphonon and multiple-scattering processescome into play, leading to a cross section σmph, which increases with energy andtemperature.
The third contribution, σbragg, arises from Bragg scattering in single- orpolycrystalline material. At low energies, below the Bragg cutoff (λ > 2dmax),σbragg is zero. In polycrystalline materials the cross section rises steeply above theBragg cutoff and oscillates with increasing energy as more reflections come intoplay. At higher energies, σbragg decreases to zero.
In single-crystalline material above the Bragg cutoff, σbragg is characterized by
3.2-12
a discrete spectrum of peaks whose heights and widths depend on the beamcollimation, energy resolution, and the perfection and orientation of the crystal.Hence, a monocrystalline filter has to be tuned by careful orientation. Figures 7-11show the total cross sections for common filter materials as a function of energy,while Fig. 12 shows the transmission of a pyrolitic graphite filter as a function ofincident wavelength.
Cooled, polycrystalline beryllium is frequently used as a filter for neutronswith energy less than 5 meV since there is an increase of nearly two orders ofmagnitude in the attenuation cross section for higher energies. BeO, with a Braggcutoff at approximately 4 meV, is also commonly used.
Pyrolitic graphite, being a layered material with good crystalline propertiesalong the c-direction but with random orientation perpendicular to it, liessomewhere between a polycrystal and a single crystal as far as its attenuation crosssection is concerned. Pyrolitic graphite serves as an efficient second- or third-orderfilter [20] and can be “tuned” by slight misorientation away from the c-axis.
Resonant absorption filters show a large increase in their attenuation cross sectionsat the resonant energy and are therefore used as selective filters for that energy. A list oftypical filter materials and their resonance energies is given in Table 3.
3.2-13
Table 3: Characteristics of some typical elements and isotopes used as neutron filters.
3.2-14
Figure 7: Total cross section for beryllium in an energy range where it can beused as a filter [19].
3.2-15
Figure 8: Total cross section for sapphire in an energy range where it can beused as a filter [19].
3.2-16
Figure 9: Total cross section for silicon in an energy range where it can be usedas a filter [19].
3.2-17
Figure 10: Total cross section for quartz in an energy range where it can beused as a filter [19].
3.2-18
Figure 11: Total cross section for pyrolitic graphite in an energy range whereit can be used as a filter [19].
Refractive lensesThe real part of the decrement of the refractive index given by
is small and positive for most materials. Hence, concave lenses must begenerally used to focus neutron beams, and the focal lengths , where R is theon-axis radius of curvature, are prohibitively long. Reducing R, to reduce the focallength to practical values, would severely limit the lens aperture.
A series of N-aligned lenses with a negligible distance between each lens has afocal length .
The factor reduction in focal length of a compound refractive lens allowsfor a reasonable value for the radius, R, of each element. Spherical surfaces makeimperfect lenses, and only the central paraxial region of these approximate theparaboloid of revolution surface of ideal lenses. For lenses with paraboloidsurfaces, R is the radius of curvature on axis at the apex of the paraboloid.
A suitable lens material should have a large value of δ and a small value of thelinear attenuation coefficient, µ. The figure of merit, , is listed for severalmaterials in Table 4. µ
δ
Ν1
βin −∂−= 1
3.2-19
Table 4: Density, linear absorption coefficient, and figure of merit of selectedmaterials at a wavelength of 1.8 Å.
cohbN.
2
2
πλ=∂
δ2R
f =
δNR
f2
=
µ δ δ/µ [m]
PolarizersMethods used to polarise a neutron beam are varied, and the choice of the besttechnique depends on the instrument and the experiment to be performed. The mainparameter that has to be considered when describing the effectiveness of a givenpolariser is the polarising efficiency, defined as
P = (N+ - N-)/(N+ + N-) , (12)
where N+ and N- are the numbers of neutrons with spin parallel (+) or antiparallel(-) to the guide field in the outgoing beam. The second important factor,transmission of the wanted spin state, depends on various factors such asacceptance angles, reflection, and absorption.
Single-crystal polarizersThe principle by which ferromagnetic single crystals are used to polarize andmonochromate a neutron beam simultaneously is shown in Fig. 13. A field B,applied perpendicular to the scattering vector κκ, saturates the atomic moments Malong the field direction B. The cross section for Bragg reflection in this geometry is
(dσ/dΩ) = FN(κκ)2 + 2 FN(κκ) FM(κκ)(P•µµ) + FM(κκ)2 , (13)
3.2-20
Figure 12: Measured transmission of a typical pyrolytic graphite filter, 4 cmthick, as a function of neutron wavelength, for neutrons traveling approximatelynormal to the (0002) graphite planes. The filter orientation is “tuned” to achieveminimum transmission of unwanted higher-order neutrons.
where FN(κκ) and FM(κκ) are the nuclear and magnetic structure factors, respectively.The vector P describes the polarization of the incoming neutron with respect to B;P = 1 for (+) spins, P = -1 for (-) spins, and µµ is a unit vector in the direction ofthe atomic magnetic moments.
Hence, for neutrons polarized parallel to B (P•µµ = 1), the diffracted intensity isproportional to [FN(κ)+FM(κ)]2, while, for neutrons polarized antiparallel to B(P•µµ = -1), the diffracted intensity is proportional to [FN(κ) - FM(κ)]2.Thepolarising efficiency of the diffracted beam is then:
P = ±2 FN(κ) FM(κ) / [FN(κ)2 - FM(κ)]2, (14)
which can be either positive or negative and has a maximum value for |FN(κκ)| = |FM(κκ)|.Thus, a good single-crystal polariser, in addition to possessing a convenient
crystallographic structure, must be ferromagnetic at room temperature and shouldcontain atoms with large magnetic moments. Furthermore, large single crystals with“controllable” mosaic should be available. Finally, the structure factor for the requiredreflection should be high, while those for higher-order reflections should be low.
None of the three naturally occurring ferromagnetic elements (Fe, Ni, and Co)makes efficient single-crystal polarizers. Cobolt is strongly absorbing, and the nuclearscattering lengths of iron and nickel are too large to be balanced by their weak magneticmoments. An exception is 57Fe, which has a rather low nuclear scattering length, andstructure-factor matching can be achieved by mixing 57Fe with Fe and 3% Si [21].
3.2-21
Figure 13: Geometry of a polarizing monochromator showing the latticeplanes (hkl) with |FN| = |FM|, the direction of P and µ, the expected spindirection and intensity.
In general, to facilitatestructure factor matching, alloysare used rather than elements. Thecharacteristics of some alloys usedas polarizing monochromators arepresented in Table 5. At shortwavelengths, the 200 reflection ofCo0.92Fe0.08 is used to give apositively polarized beam [FN(κ)and FM(κ) both positive], but theabsorption from cobalt is high. Atlonger wavelengths, the (111)reflection of the Heusler alloyCu2MnAl [22] is commonly usedsince it has a higher reflectivity(Fig. 14) and a larger d-spacingthan Co0.92Fe0.08. Since, for the(111) reflection, FN ~ -FM, thediffracted beam is negatively polarized. Unfortunately, the structure factor of the(222) reflection is higher than that of the (111) reflection, leading to significanthigher-order contamination of the beam.
3.2-22
Table 5: Properties of polarizing crystal monochromators.
Figure 14: Calculated and measuredreflectivity from the (111) reflection of aHeusler crystal with an intrinsic mosaicof 0.045°.
Polarizing mirrorsFor a ferromagnetic material, the neutron refractive index is given by
= 1 - λ2 N(bcoh ± p)/2π , (15)
where, the magnetic scattering length, p, is defined by
p = 2µ(B – H) mπ/h2N , (16)
where, m and µ are the neutron mass and magnetic moment, B is the magneticinduction in an applied field H, and h is Planck’s constant.
The - and + signs refer, respectively, to neutrons whose moments are alignedparallel and antiparallel to B. The refractive index depends on the orientation of theneutron spin with respect to the film magnetization, thus giving rise to two criticalangles of total reflection, γ- and γ+. Thus, reflection in an angular range betweenthese two critical angles gives rise to polarized beams in reflection and intransmission. The polarization efficiency, P, is defined in terms of the reflectivityr+ and r- of the two spin states:
P = (r+ - r-)/(r+ + r-) . (17)
For optimum reflectivity, polarizing mirrors are usually made by depositing thinfilms of ferromagnetic materials onto substrates of low surface roughness (e.g., floatglass or polished silicon). The reflection from the substrate can be reduced byincluding an antireflecting layer made from, for example, gadolirium/titanium alloys.
The major limitation of such polarizers is that grazing-incidence angles mustbe used and the angular range of polarization is small. This limitation can bepartially overcome by using multilayers, as described previously, in which one ofthe layer materials is ferromagnetic. In this case the refractive index of theferromagnetic material is matched for one spin state to that of the nonmagneticmaterial so that reflection does not occur. A polarizing supermirror made in thisway has an extended angular range of polarization (Figs. 15 and 16).
Note that modern deposition techniques allow the refractive index to beadjusted readily so that matching is easily achieved. The scattering-length densitiesof some commonly used layer pairs are given in Table 6.
Polarizing multilayers are also used in monochromators and broadbanddevices. Depending on the application, various layer pairs can be used:Co/Ti, Fe/Ag, Fe/Si, Fe/Ge, Fe/W, Fe50Co48V2/TiN, FeCoV/TiZr, and63Ni0.66
54Fe0.34/V, and the range of fields used to achieve saturation varies fromabout 10 to 500 Gs.
Polarizing mirrors can be used in reflection or transmission with polarization
n±
3.2-23
3.2-24
Figure 15: Measured performance of an m = 3 Fe/Si polarizing supermirrorin reflection geometry. The reduction in reflectivity at low values of m is anartefact due to underillumination of the sample during the measurement.
Figure 16: Measured performance of an m = 3 Fe/Si polarizing supermirror intransmission geometry. The increase in transmission at low values of m is anartifact caused by underillumination of the sample during the measurement.
efficiencies reaching 97%, although because of the low incidence angles, their useis generally restricted to wavelengths above 2 Å. Various devices can beconstructed using mirror polarizers including simple reflecting mirrors, V-shapedtransmission polarizers [23], cavity polarizers [24], and benders [25].
Polarizing FiltersPolarizing filters operate by selectively removing one of the neutron spin statesfrom an incident beam, allowing the other spin state to be transmitted with only
3.2-25
Table 6: Scattering-length densities for some typical materials used forpolarizing multilayers. For the nonmagnetic layer, we have listed only thesimple elements that give a close match to the N(b-p) value of thecorresponding magnetic layer. In practice, excellent matching can be achievedby using alloys (e.g., TixZry alloys allow niobium b values between -1.95 and3.03 x 10-6 Å-2 to be selected) or reactive sputtering (e.g. TiNx).
moderate attenuation. The spin selection is obtained by preferential absorption orscattering, so the polarizing efficiency usually increases with the thickness of thefilter, whereas the transmission decreases. A compromise must therefore be madebetween polarization, P, and transmission, T. The “quality factor” often used isP [26].
The total cross sections for a generalized filter may be written as
σ± = σ0 ± σp , (18)
where σ0 is a spin-independent cross section and σp = (σ+ + σ-)/2 is the polarizationcross section. It can be shown [27] that the ratio σp/σ0 must be ≥ 0.65 to achieve |P|> 0.95 and T > 0.2.
Magnetized iron was the first polarizing filter to be used. The method relies onspin-dependent Bragg scattering from a magnetized polycrystalline block for whichσp approaches 10 barns near the iron cutoff at 4 Å. For wavelengths in the range 3.6to 4 Å, the ratio σp/σ0 is ~ 0.59, resulting in the polarizing efficiency of 0.8 for atransmittance of ~ 0.3. In practice, however, since iron cannot be fully saturated,depolarization occurs, and values of P ~ 0.5 with T ~ 0.25 are more typical.
Resonance absorption polarization filters rely on the spin-dependence of theabsorption cross section of polarized nuclei at their nuclear resonance energy and canproduce efficient polarization over a wide energy range. The nuclear polarization isnormally achieved by cooling in a magnetic field, and filters based on 149Sm (Er =0.097 eV) and 151Eu (Er = 0.32 eV and 0.46 eV) have been tested successfully [28].The 149Sm filter has a polarizing efficiency close to 1 within a small-wavelengthrange (0.85 to 1.1 Å), while the transmittance is about 0.15.
Broadband polarizing filters, based on spin-dependent scattering orabsorption, provide an interesting alternative to polarizing mirrors ormonochromators because of the wider range of energy and scattering angle that canbe accepted. The most promising such filter is polarized 3He, which operatesthrough the huge spin-dependent neutron capture cross section that is totallydominated by the resonance capture of neutrons with antiparallel spin. Thepolarization efficiency of a 3He neutron spin filter of length, l, can be written as
Pn(λ) = tanh[O(λ)PHe] , (19)
where PHe is the 3He nuclear polarization and O(λ) = [3He] l σ0(λ) is thedimensionless effective absorption coefficient, also called the opacity [29]. Forgaseous 3He, the opacity can be written in more convenient units by
O’ = p[bar] x l[cm] x λ[Å] , (20)
T
3.2-26
where p is the 3He pressure and O =7.33 x 10-2 O.’Similarly, the residual transmission of the spin filter is given by:
(21)
It can be seen from Fig. 17, that even at low 3He polarization, full neutronpolarization can be achieved in the limit of large absorption at the cost of thetransmission.
3He can be polarised either by spin exchange with optically pumped Rb [30] or bypumping of metastable 3He* atoms followed by metastable exchange collisions [31]. Inthe former method, the 3He gas is polarized at the required high pressure whereas 3He*
pumping takes place at a pressure of about 1 mbar, followed by a polarization conserv-ing compression by a factor of nearly 10,000. Although the polarization time constantfor Rb pumping is of the order of several hours, compared with fractions of a second for3He*, pumping the latter requires several “fills” of the filter cell to achieve the requiredpressure. Typically, nuclear polarizations PHe of 55% are achieved.
Zeeman PolarizerThe reflection width of perfect silicon crystals for thermal neutrons and the Zeemansplitting (∆Ε = 2µB) of a field of about 10 kG are comparable and therefore can beused to polarize a neutron beam. For a monochromatic beam (energy E0 ) in a strongmagnetic field region, the result of the Zeeman splitting will be a separation into twopolarized subbeams, one polarized along B with energy E0 + µB and the otherpolarized antiparallel to B with energy E0 - µB. The two polarized beams can beselected by rocking a perfect crystal in the field region B [32].
Spin-orientation devicesPolarization is the state of spin orientation of an assembly of particles in a target orbeam. The beam polarization vector P is defined as the vector average of this spinstate of the assembly in the beam, often described by the density matrixρ = (1+σσP). The polarization is then defined as P = Tr(ρρσσ). If the polarizationvector is inclined to the field direction in a homogenous magnetic field, B, thepolarization vector will precess with the classical Larmor frequency ωL = |γ|B.This results in a precessing vector and a precessing spin polarization. For mostexperiments, it is sufficient to consider the linear polarisation vector in the directionof an applied magnetic field. If, however, the magnetic field direction changesalong the path of the neutron, it is also possible that the direction of P will change.If the frequency, Ω, with which the magnetic field changes is such that
Ω = d(B/|B|)/dt << ωL , (22)
1
2
T = exp O cosh On λ λ λ( ) − ( )[ ] ( ) ⋅[ ]PHe
3.2-27
then the polarization vector follows the field rotation adiabatically. Alternatively,when Ω >> ωL , the magnetic field changes so rapidly that P cannot follow and thecondition is known as nonadiabatic fast passage. All spin-orientation devices arebased on these concepts.
Maintaining the direction of polarizationA polarised beam will tend to become depolarized during passage through a regionof zero field since the field direction is ill defined over the beam cross section.Thus, to keep the polarization direction aligned along a defined quantization axis,special precautions must be taken.
The simplest way of maintaining the polarization of neutrons is to use a guidefield to produce a well-defined field B over the whole flight path of the beam. If thefield changes direction, it has to fulfill the adiabatic condition Ω << ωL, that is thefield changes must take place over a time interval that is long compared with theLarmor period. In this case, the polarization adiabatically follows the field directionwith a maximum angle of deviation, ∆Θ ≤ 2 tan-1(ω/ωL) [33].
3.2-28
Figure 17: Neutron polarization and transmission of a 3He filter with 55%nuclear polarization. Also shown is the quality factor P .T
Rotation of the polarization directionThe polarization direction can be changed by the adiabatic change of the guide fielddirection so that the direction of the polarization follows it. Such a rotation isperformed by a spin turner or spin rotator [34].
Alternatively, the direction of polarization can be rotated relative to the guidefield by using the property of precession described previously. If a polarized beamenters a region where the field is inclined to the polarization axis, then thepolarization vector P will precess about the new field direction. The precessionangle will depend on the magnitude of the field and the time spent in the fieldregion. By adjustment of these two parameters together with the field direction, adefined, though wavelength-dependent, rotation of P can be achieved. A simpledevice uses the nonadiabatic fast passage through the windings of two rectangularsolenoids wound orthogonally on top of each other [35]. In this way the mechanicalrotation can be replaced by two currents the ratio of which defines the direction ofthe precession field axis and the size of the fields determines the angle φ of theprecession. The orientation of the polarization vector can therefore be defined inany direction.
In order to produce a continuous rotation of the polarization (i.e., awell-defined precession) as, required in neutron spin echo applications, precessioncoils are used. In the simplest case these are long solenoids for which the change ofthe field integral over the cross section can be corrected by Fresnel coils [36].Zeyen and coworkers have developed and implemented optimal field shape coils(ofs) [37]. The field in these coils follows a cos2 shape that results from theoptimization of the line integral homogeneity. The OFS coils can be wound over avery small diameter, drastically reducing stray fields.
Flipping of the polarization directionThe term “flipping” was originally applied to the situation where the beampolarization direction is reversed with respect to a guide field; that is. it describes atransition of the polarization direction from parallel to antiparallel to the guide fieldand vice versa. A device that produces this 180° rotation is called a π-flipper. A π/2flipper, as the name suggests, produces a 90° rotation and is normally used toinitiate precession by turning the polarization at 90° to the guide field.
The most direct, wavelength-independent way of producing such a transitionis again a nonadiabatic fast passage from the region of one field direction to theregion of the other field direction. This can be realized by a current sheet like theDabbs foil [38], a Kjeller eight [39], or a cryoflipper [40].
Alternatively, a spin flip can be produced using a precession coil, as described
3.2-29
previously, in which the polarization direction makes a precession of just π about adirection orthogonal to the guide field direction. Normally, two orthogonallywound coils are used, where the second, correction coil, serves to compensate theguide field in the interior of the precession coil. Such a flipper is wavelengthdependent and can be easily tuned by varying the current in the coils.
Mechanical Choppers and SelectorsIn this section we deal with devices that take advantage of the neutron flight time tomake a selection of some sort. To put things into perspective, we should mention that a4-Å neutron has a reciprocal velocity of approximately 1,000 µs m-1 so that accurateneutron energy determinations can be made with flight paths of only a few meters.
Disc choppers rotating at speeds up to 20,000 rpm about an axis that is parallelto the neutron beam are used to produce a well-defined pulse of neutrons. The discsare made from absorbing material (at least where the beam passes) and compriseone or more neutron-transparent apertures or slits. For polarized neutrons thesetransparent slits should not be metallic, as the eddy currents in the metal moving ineven a weak guide field will strongly depolarize the beam. The pulse frequency isdetermined by the number of apertures and the rotation frequency, while the dutycycle is given by the ratio of open time to closed time in one rotation. Two suchchoppers rotating in phase can be used to monochromate and pulse a beamsimultaneously [41]. In practice, more than two choppers are generally used toavoid frame overlap of the incident and scattered beams. The time resolution ofdisc choppers (and hence the energy resolution of the instrument) is determined bythe beam size, the aperture size, and the rotation speed. For a realistic beam size therotation speed limits the resolution.
Therefore, in modern instruments it is normal to replace a single chopper withtwo counter-rotating choppers [42]. The low duty cycle of a simple disk choppercan be improved by replacing the single slit with a series of slits either in a regularsequence (Fourier chopper) [43] or a pseudostatistical sequence (pseudostatisticalchopper) [44] with duty cycles of 50% and 30%, respectively.
The Fermi chopper is an alternative form of neutron chopper thatsimultaneously pulses and monochromates the incoming beam. It consists of a slitpackage, essentially a collimator, rotating about an axis that is perpendicular to thebeam direction [45]. For optimum transmission at the required wavelength, the slitsare usually curved to provide a straight collimator in the neutron frame ofreference. The curvature also eliminates the “reverse burst,” that is. a pulse ofneutrons that passes when the chopper has rotated by 180°.
A Fermi chopper with straight slits in combination with a monochromator
3.2-30
assembly of wide horizontal divergence can be used to time focus a polychromaticbeam, hence maintaining the energy resolution while improving the intensity [46].
Velocity selectors are used where a continuous beam is required with coarseenergy resolution. They exist in either multiple disc configurations or helicalchannels rotating about an axis parallel to the beam direction [47]. Modern helicalchannel selectors are made up of lightweight absorbing blades slotted into helicalgrooves on the rotation axis [48]. At higher energies, where no suitable absorbingmaterial is available, highly scattering polymers (polymethyl metacrylate) can beused for the blades, although in this case adequate shielding must be provided. Theneutron wavelength is determined by the rotation speed and resolutions, ∆λ/λ;range from 5 to practically 100% (λ/2 filter) can be achieved. The resolution isfixed by the geometry of the device but can be slightly improved by tilting therotation axis or can be relaxed by rotating in the reverse direction for shorterwavelengths. Nowadays transmissions of up to 94% are obtainable.
References[1] Anderson, I. S., Brown, P.J., Carpenter, J.M., Lander, G., Pynn, R., Rowe, J.M.,Schärpf, O., Sears, V.F., and Willis, B.T.M., Intern. Tables for Crystallography C,2nd Edition, Kluwer Academic Publish. (1999) 426
[2] Nunes, A. C., NIM 119 (1974) 291; Glinka, C. J., Rowe, J. M., and LaRock, J.G., J. Appl. Cryst. 19 (1986) 427
[3] Wright, A. F., Berneron, M., and Heathman, S. P., NIM 180 (1981) 655-658
[4] Maier-Leibnitz, H., Summer School on Neutron Physics, Alushta Joint Insituteof Nuclear Physics (Russia) (1969)
[5] Bonse, U., and Hart M., Appl. Phys. Lett. 7 (1965) 238-240
[6] Zachariasen, W. H., “Theory of X-ray Diffraction in Crystals,” Dover, NewYork (1945)
[7] Sears, V. F., Acta Cryst. A53 (1997) 35-36
[8] Alefed, B., Kerntechnik 14 (1972) 15-17
[9] Hock, R., Vogt, T., Kulda, J., Mursic, Z., Fuess, H., and Magerl A., Z. Phys B 90(1993) 143.
[10] Freund, A. K., Guinet, P., Mareschal, J., Rustichelli, F., and Vanoni, F., J.Cryst. Growth 13-14 (1972) 726
[11] Magerl, A., Liss, K. D., Doll, C., Madar R., and Steichele, E., NIM A 338 (1994) 83-89
[12] Riste, T., NIM 86 (1970) 1-4; Currat, R., NIM 107 (1973) 21-28
3.2-31
[13] Scherm, R., Dolling, G., Ritter, R., Schedler, E., Teuchert, W., and Wagner, V.,NIM 143 (1977) 77-85
[14] Scherm, R. H., and Kruger, E., NIM A 338 (1994) 1
[15] Christ, J., and Spinger, T., Nukleonik 4 (1962) 23-25
[16] Schoenborn, B. P., Caspar, D. L. D., and Kammerer, O. F., J. Appl. Cryst. 7(1974) 508-510
[17] Mezei, F., Commun. Phys. 1 (1976) 81-85; Hayter J. B. and Mook H. A., J.Appl. Cryst. 22 (1989) 35-41
[18] Mildner, D.F.R., Chen-Mayer, H.H., and Downing, R.G., J. Phys. Soc. Jpn 65(1996) Suppl. A, 308-311
[19] Freund, A. K., NIM 213 (1983) 495-501
[20] Shapiro, S. M., and Chesser, N. J., NIM 101 (1972) 183-186
[21] Reed, R. E., Bolling, E. D., and Harmon, H. E., Solid State Division Report.Oak Ridge National Laboratory, USA (1973) 129
[22] Delapalme, A., Schweizer, J., Couderchon, G., and Perrier de la Bathie, R.,NIM 95 (1971) 589-594; Freund, A., Pynn, R., Stirling, W. G., and Zeyen, C. M. E.,Physica B 120 (1983) 86-90
[23] Majkrzak, C. F., Nunez, V., Copley, J. R. D., Ankner, J. F., and Greene, G. C.in “Neutron optical devices and applications,” edited by C. F. Majkrzak and J. L.Wood., SPIE Proc. 1738 (1992) 90-106
[24] Mezei, F. in “Thin film neutron optical devices: mirrors, supermirrors,multiplayer monochromators, polarizers and beam guides,” edited by C. F.Majkrzak., SPIE Proc. 983 (1988) 10-17
[25] Schärpf, O., Physica (Utrecht) B 156 & 157 (1989) 639-646
[26] Tassel, F. and Resouche, E., NIM A 359 (1995) 537-541
[27] Williams, W. G, “Polarized Neutrons,” Claredon Press, Oxford (1988)
[28] Freeman, F. F., and Williams, W. G., J. Phys. E. 11 (1978) 459-467
[29] Surkau, R., Becker, J., Ebert, M., Grossman, T., Heil, W., Hofmann, D.,Humblot, H., Leduc, M., Otten, E. W., Rohe, D., Siemensmeyer, K., Steiner, M.,Tasset, F., and Trautmann N., NIM A 384 (1997) 444-450
[30] Bouchiat, M., A., Carver, T., R., and Varnum, C., M., Phys. Rev. Lett. 5 (1960)373-375; Chupp, T., E., Coulter, K., P., Hwang, S., R., Smith, T., B., and Welsh, R.,C., J., J. Neutron Res. 5 (1996) 11-24
3.2-32
[31] Colgrove, F., D., Schearer, L., D., and Walters, K., Phys. Rev. 132 (1963)2561-2572
[32] Forte M. and Zeyen C. M. E., NIM A 284 (1989) 147-150
[33] Schärpf, O., in “Neutron spin echo,” Lecture notes in physics, vol. 128, editedby F. Mezei, Springer-Verlag (1980) 27-52
[34] Schärpf, O., and Capellman, H., Phys. Status Solidi A 135 (1993) 359-379
[35] Jones, T. J. L., and Williams, W. G., Rutherford Laboratory Report RL-77-O79/A(1977)
[36] Mezei, F., Z. Phys. 255 (1972) 146-160
[37] Zeyen, C. M. E., and Rem, P. C., Meas. Sci. Tech. 7 (1996) 782
[38] Dabbs, J. W. T., Roberts, L. D., and Bernstein, S., Report ORNL-CF-55-5-126.Oak Ridge National Laoratory, USA (1955)
[39] Abrahams, K., Steinsvoll, O., Bongaarts, P. J. M., and De Lange, P. W., Rev.Sci. Instrum. 33 (1962) 524-529
[40] Forsyth, J. B., At. Energy Rev. 17 (1979) 345-412
[41] Egelstaff, P. A., Cocking, S. J., and Alexander, T. K., Inelastic Scattering ofneutrons in solids and liquids. IAEA, Vienna (1961) 165-177
[42] Hautecler, S., Legrande, E., Vansteelandt, L., d’Hooghe, P., Rooms, G.,Seeger, A., Schalt, W., and Gobert, G., Proccedings of the Conference on “NeutronScattering in the Nineties,” Jülich. IAEA, (1985) 211-215; Copley J. R. D., NIM A303 (1991) 332-341
[43] Colwell, J. F., Miller, P. H., and Whittemore, W. L., “Neutron InelasticScattering”, Vol. II, IAEA, Vienna (1968) 429-437
[44] Hossfeld, F., Amadori, R., and Scherm, R., “Instrumentation for NeutronInelastic Scattering,” IAEA, Vienna (1970) 117
[45] Turchin, V. F., “Slow neutrons,” Israel Programme for Scientific Translation(1965)
[46] Blanc, Y., “Le spectrometre à temps de vol IN6: characteristiques, techniqueset performances.” ILL internal Report No 82BL21G, Institut Laue-Langevin,Grenoble (1983)
[47] Dash, J. G., and Sommers, H. S., Rev. Sci. Instrum. 24 (1953) 91-96
[48] Wagner, V., Friedrich, H., Wille, P., Physica B 180&181 (1992) 938-940
3.2-33
Detectors for Thermal Neutrons
A.Oed
IntroductionThe particle- or radiation-detection is based on the measurement of electric currents. Thermal neutrons with their low velocity and without an electric charge can onlybe measured subsequent to a nuclear reaction with target atoms which emit eitherionizing radiations or ionizing particles. The target isotopes commonly used forthermal neutron detection are indicated in Tab (1). Except for Gd, all the otherreactions are fission processes where two fission particles are ejected in oppositedirections randomly oriented in space.The Maxwell distributions of thermal neutrons are shown in Fig (1).
3.3-1
Table 1: Commonly used isotopes for thermal neutron detection, reactionproducts and their kinetic energies.
Figure 1: Velocity-, energy- and wave-length-distribution for neutrons at 300 o K.
Absorption lawA neutron flux Jo [1/s] after having passed the length of x [cm] in an absorberwith the absorption length µ [cm], is reduced to the value
J= Jo e - (x/µ)
0r the relative amount of the flux which disappears in the absorber is
(Jo-J) / Jo = 1- e - (x/µ)
Expressed in percentage, this leads to the efficiency of the detector or absorber
Eff = (1- e -( x/µ))* 100 The absorption length µ [cm] is inversely proportional to the product of the cross section σ[cm2] and the atomic density Ad [atom/cm3] of the absorber
µ = 1/( σ*Ad)
The absorption law expressed with the cross-section therefore is
J= Jo e -( σ* x* Ad)
3.3-2
Table 2: Cross-section, absorption length and mass-absorption–density forthermal neutrons. (v = 2224 m/s; λλ = 1.78 Å ; E kin = 26 m eV ; T = 300 °K).Isotopes with high cross-section and therefore used in neutron detection aremarked in bold type.
For gases at 1 bar, the atomic density Ad is 2.7 1019 [1/cm3] and for solids and liquidsits value is about 1000 times higher. In general, the atomic density is given by
Ad = ρ * Na / Mv [1/cm3] where
ρ is the volumetric weight [ g/cm3], Na= 6.25 1023 the Avogadro number[atom/ mol ] and Mv the molar weight of the absorber-material [g /mol].
Sometimes the absorption is also expressed in terms of the mass-absorption-coefficient γ = µ * ρ [ g / cm2]. With the surface mass density Md =x*ρ [g/cm2] ofan absorber of x [cm] in thickness, the law reads:
J= Jo e –(Md / γ )
The absorption length, cross section and mass absorption-coefficient for thermalneutrons for some materials used in neutron experiments are indicated in Tab (2)
The cross-section σ for thermal neutrons is inversely proportional to their velocity v.
where E is kinetic energyσ σ ∼ / ∼ 1 / 1 v or E
3.3-3
Figure 2: Neutron cross-section as function of the kinetic energy for theisotopes commonly used in thermal neutron detection.
As the Broglie wavelength λ = h / (mn • v) with h = Planck–constant andmn= neutron mass is also inversely proportional to the velocity v, the cross-sectionincreases linearly with the neutron wavelength λ: σ ~ λ.157Gd and 235 U have resonances at higher energies, as shown in Fig (2)
Neutron detectorsThe following list is a survey of all possible neutron detectors subdivided into integratingand counting devices. Its characteristics are indicated as advantages and disadvantages.
Without taking into account the importance of the different qualities and especiallythe costs, it seems that counting detectors have more advantages than the integratingones. Integrating detectors are more suitable for beam monitoring, beam alignment andneutronography whereas the counting devices are appropriate in interference patternrecordings for structure analysis and time-depending changes inside the samples.
3.3-4
COUNTINGConverter foil with
pixel semiconductor;A1,A2,A3,A4,A5, D6,D8
pos. sensitive. gas counter;A3,A4A5,A6,A8, D1,D2
n-Scintillator withpos. sensitive PM;A1,A2,A3,A4,A5,A6, A7,D8
pos. sensitive gas counter;A3,A4,A5,A6,A7, A8, D1,D2
Internal convertergas-counter, -array;A3,A4,A5,A6,A7,A8, D1,D2
multi wire prop. counter.;A3,A4,A5,A6,A7,A8, D1,D2
micro pattern gas counter;A1,A3,A4,A5,A6,A7, A8, D2
boron diode;A1,A2,A3,A4,A5,A6,A7, D8
INTEGRATINGConverter foil with
photogr. film; A1,A2,A8, D3,D4,D5,D6,D7
image plate; A1,A2,A5,A8, D3,D4,D6,D7
CCD camera; A1,A2,A5, D3,D4,D6,D7,D8
n-Scintillator withCCD camera;A1,A2,A5, D3,D4,D7
Thomson tube;A1,A2,A5,A6,A8, D3,D4,D7
Image amplifier;A1,D2,D3,D5,D6,D7,D8
Internal converterionization chamber;A1,A4,A5,A8, D2,D3
Gd loaded image plate;A1,A2,A5,A8, D3, D4,D7
Gas-detectorsIonization chamberThe simplest neutron gas detector is the ionization chamber: a condenser consistingof two metallic plates and the gap in between filled with 3 He or 10 BF3 gas.
The two particles emitted by the nuclear reaction are slowed down in the gas.Along their trace they produce electron-ion pairs, and with an electric field strengthof about 100 V/(cm . bar) these charges are collected on the corresponding plateswhere the current can be measured. To generate one electron-ion pair in a gas, theionizing particle loses about 30 eV of its energy. This is why in the n (3He, p) 3H+0.77 MeV reaction for one neutron about 22 000 electron-ion pairs are producedand a charge of 4.1 10-15 [As] is collected. Usually ionization chambers are used asbeam monitors.
The triton range is about 1/3 of the proton range as can be seen in Fig (4). Thismeans that there exists a systematic deviation between the centre of gravity of the
3.3-5
AdvantagesA1) high intensity capabilityA2) high position resolutionA3) gamma discriminationA4) on-line read-out, short time slicesA5) high dynamicA6) high n-efficiencyA7) low noiseA8) large sensitive area
DisadvantagesD1) limited counting capacity < 1 kHz/mm2
D2) limited position resolution > 100 µcr.D3) gamma sensitivity D4) long read-out time > 1ms /frameD5) low dynamic < 1/100 D6) low n-efficiency < 20 %D7) high noise > 1 count / pixelD8) small sensitive area < 100 cm 2
Some useful constants:
k= 1.384 10-23 Joule/ °K = kg(m/s)2/ °K Boltzmann constant
h= 6.6252 10-34 VAs = Joule*s= kg (m/s)2 s Planck constant
e= 1.62 10-19 As elementary charge
v= 2.998 108 m/s velocity of light
me= 9.108 10-31 kg electron mass
mp= 1.6723 10-27 kg proton mass
mn= 1.6747 10-27 kg neutron mass
Na= 6.25 1023 atom/Mol Avogadro number
charges and the location where the reaction takes place.This deviation can be reduced be adding other gases with higher molecular
weight, preferably composed of light atoms like C, H and F in order to be lesssensitive to x- rays and gammas. For some gases, the mean energy loss of a particleto produce one electron-ion pair, the particle range and the mean deviation from thereaction point are indicated in Tab (3).
Proportional counterThe small amount of charges in an ionization chamber (see above) produced by asingle neutron is difficult to measure, whereas this is easy to do with a Geigercounter 2. A very thin wire of several microns in diameter , which serves asanode, is located in the axe of a cylindrical tube. This condenser is filled with gas.The electrons produced by the ionizing particles of the nuclear reaction drifttowards the positive wire where they will experience an avalanche amplification atthe very high field strength surrounding the wire. This gas amplification isnoiseless and proportional to the amount of primarily produced charges. The gasgain reaches factors up to 105. Single neutrons are easily detectable. A typical pulseheight spectrum of a 3He +Ar +CH4 counter is shown in Fig (5).
As a matter of fact, at 770 keV only one line would be expected,corresponding to the total reaction energy. But some of the particle traces enter intothe housing and the charges are lost. At the point of 192 keV the reaction has taken
3.3-6
Figure 3: Neutron detection efficiency as a function of the detection gap filledwith 1 bar 3 He.
place directly on the inner surface of the cylinder and the proton is stopped in themetal. Only the triton, emitted in the opposite direction, releases its completeenergy into the gas. With a higher gas pressure the amount of counts in the tail willbe smaller as the range of the particles will be shorter. Between the neutron signalsand the gamma background there is a large gap which allows a very good gammadiscrimination. Unfortunately the local counting capacity is limited to about1 kHz / mm2. The ions generated in the gas amplification very slowly drift towards
3.3-7
Figure 4: Ionization /mm in He of the particles emitted by the reactionn (3He, p) 3H +0.77 MeV. Ionization and Range are calculated with theProgram :TRIM86 1.
Table 3: Mean ionization energy, particle range and deviation in some gases
the cylinder walls. At high rates their space charge reduces the applied electric fieldon the wire and hence the gas gain. For a good energy resolution, which allows agood gamma discrimination, all the primary charges of one event must becollected. As the ionizing particles are emitted in random directions with respect tothe wire orientation, the drift time of the electrons towards the wire variesconsiderably due to the inhomogeneous cylindrical field. The drift time depends onthe gas mixture and on the local electric field strength. In general, up to 10 µs arenecessary for a good charge collection. The maximal count rate of the wholecounter therefore is of about 30 kHz with a pileup of 30% of the signals. The driftvelocity of electrons in a CF4– He mixture is shown in Tab (4). With He the highestdrift velocity is reached by adding CF4.
As shown recently 4, a counter filled with N2 +CF4 is an excellent neutronbeam monitor. For a 1 cm gap filled with 1 bar N2 the neutron efficiency is9.05 10-5 at a wavelength of 1.8 Å. In the reaction n (14N, 14C) H + 627 keV, with across-section of 1.8 barn , the proton is emitted with an energy of 585 keV anddelivers a charge signal which is very well separated from the gamma background.
3.3-8
Figure 5: Pulse height spectrum of a 3He neutron counter irradiated with alab. Am–Be source. In order to distinguish between gamma background andneutron signal, a boron loaded plastic absorber was installed between sourceand counter in a second measurement.
Multi wire proportional chamber (MWPC)The position sensitive variant of the Geiger counter is the MWPC invented byCharpak5.
It consists of a great number of thin anode wires assembled in a plane andmounted between two plates which are composed of separated cathode strips. Theorientation of the strips is orthogonally to each other. As with Geiger counters, thegas amplification takes place on the wires. After the separation of the electron-ionpairs in the avalanche by the electric field, the ion cloud influences signals on thecathode strips which allows to determine the position of the event. The achievableposition resolution perpendicular to the wires depends on the wire spacing.
With a wire spacing smaller than one millimeter the operation of this device isextremely difficult. As the electric field between the anode frame and the cathodes ishomogeneous, the distribution of the drift time of the primary charges is more uniformcompared to a cylindrical counter and therefore the time for the charge-collection isreduced to about 1 µs. As the counting capacity only depends on this collection time,the rate limit for a two-dimensional detector increases to about 300 kHz for the wholesensitive surface, with a position error of 30 % at this rate. To record a higher rate, thedetector has to be segmented. A one dimensional detector with individual read-out pres-ents such a segmentation. Each cell is able to record the rate of 300 kHz. In any casethe local counting capacity is limited to about 1 kHz/ mm2 just as for the counter tube.
Micro Pattern Gas Counter ( MPGC)To improve the counting capacity, the cathode distance has to be reduced for afaster evacuation of the avalanche ions; for a better position resolution, thestructure pitch of the electrodes has to be smaller. This aim has been reached withthe recently developed Micro Pattern Gas Counters. Their very small electrodestructures are manufactured by means of the photolithographic technique, acommon procedure to fabricate integrated circuits.
The first type was the Micro Strip Gas Chamber (MSGC) 6 which will beshortly described. Its arrangement corresponds to an ionization chamber but the anodeplate is made of a glass plate on the surface of which very thin conductor-strips are
3.3-9
Table 4: Drift velocity of electrons inan electric field of 1.14 V/ ( cm . mb )in a gas mixture 3.
fixed. Small strips, with a width of several microns, are located between two largerstrips in a distance of several ten microns. The applied electric potential alternatesbetween each strip. The smaller strips are the anodes, the larger ones the cathodes. Thismicro strip plate operates just like a proportional counter. An electron producedanywhere in the gas volume drifts towards the positive strip where it will experiencean avalanche amplification. But now more than 90 % of the ions produced in theavalanche are neutralised on the very close cathode strips and therefore the spacecharge is significantly reduced. A local rate limit in neutron detection of 20 kHz / mm 2
was reached , 20 times higher than in a wire counter. The overall rate limit is the sameas in a MWPC, as it only depends on the charge collection time. In neutron detectionthe smaller pitch of such a structure does not automatically lead to a better localisationbecause the precision also depends on the intrinsic deviation between the reactionpoint and the centre of gravity of the charges. The spatial resolution is only improvedif the particle range is reduced to the size of the structure pitch.
A very recent development is the Gas Electron Multiplier (GEM) 7. Itsamplifying part consists of a Polyimide foil metallized on both sides with smallholes drilled through. A potential difference between the two surfaces generates thenecessary field strength for the avalanche amplification. This device is inexpensiveand can be produced in large sizes up to 30 cm x 30 cm. A survey of the recentMPGC developments can be found in 8 and 9.
The general characteristics of gaseous neutron detectors are:1) high and noiseless internal amplification2) very good gamma discrimination3) large sensitive areas4) radiation hard5) max local rate: <= 1 kHz/ mm2 respectively < = 20 kHz/ mm2
6) total rate per cell or per sensitive surface < = 300 kHz 7) position resolution > = 1 mm
Scintillator detectors In Fig (6) the pulse height spectra of two neutron scintillators, directly coupled to aphotomultiplier, are shown. Only the glass scintillator leads to a tolerableseparation between neutron- and gamma-signals. The photon output of the lithiumloaded ZnS- powder is higher but this is also true for its gamma sensitivity.
The characteristics of some neutron scintillators are indicated in Tab (5). As a result of the higher density of the scintillator material, the range of the
ionizing particles is reduced to a few microns which can result in a better spatialresolution. The light emitted from the scintillator is either directly recorded by a
3.3-10
light sensitive detector or collected by different optical means as there are lenses,mirrors or fibres towards such a detector. This can be a photographic film, a CCDcamera, a photo multiplier (PM) or a photo-sensitive gaseous counter. But all theintegrating light-devices also record the light generated by the gammas.
On a continuous neutron beam the lithium loaded ZnS scintillator NE422 isapplicable only for measurements where its gamma sensitivity is of littleimportance for the result: in beam- monitoring or neutronography. The gammabackground can only be discriminated on a pulsed neutron beam and with acounting light detector, taking advantage of the time of flight difference betweenneutrons and gammas.
Owing to the short decay time of the glass-scintillator NE 905, a local rate up toseveral MHz / mm2 can be recorded with a directly connected PM. This is also validfor a position sensitive PM. The combination of the latter with this scintillator is theneutron detector with the highest rate capability and with a spatial resolution of about
Table 5: Characteristics of the most useful scintillators for thermal neutrons 12.
Figure 6: Pulse height spectra of the scintillators NE 905 and NE 422(Nuclear Enterprises) irradiated with a lab. Am-Be neutron source(reaction: n ( 6Li , αα ) 3H + 4.79 MeV ) and measured with a photo multiplierat the same amplification. In a second measurement a boron loaded plasticabsorber was installed between source and scintillator in order to distinguishbetween gamma background and neutron signal.
3.3-11
200 microns. The maximum sensitive area though is still limited to 8 x 8 cm2 and thecosts for this arrangement are rather high. Unfortunately the light output of theNE 905 scintillator is too low for single neutron detection with a CCD.
The Anger camera 10 is the combination of a scintillator and a light dispersercoupled to a bundle of photomultipliers. The localisation of the neutron event isachieved by determining the centroid between the different PM signals. Its sensitivearea is up to several hundred cm2 and its position resolution is of several mm.
The commercially available Thomson tube looks like an inverse cathode raytube. The screen is replaced by a Gd Oxide–scintillator, covered with aphotocathode. The light produced by the Gd-conversion electron in the scintillatorreleases an electron from the photocathode into the vacuum where it is acceleratedand focused, by an electrostatic optic, onto an output screen. There it produces ahuge amount of photons. This spot can be recorded by all the light sensitive devicesmentioned above. The sensitive area has a diameter of 215 mm and a spatialresolution of 200 to 300 microns.
For beam monitoring, a so-called “Handmonitor” 11 combines the lithiumloaded ZnS scintillator with a commercially available image amplifier. A neutronbeam profile can be directly seen on the screen of the device.
Photographic films have a dynamic of 1:100 at the most. A spatial resolutionof some hundred microns can be achieved depending on the optic and the thicknessof the scintillator. But as this is an integrating device, the light of the gammabackground is also recorded.
At present, R&D work is done for inorganic scintillators with a higher lightoutput for the future spallation neutron sources. A summary is published by CarelW.E. van Eijk 12
Foil detectorsFig (7) shows the calculated pulse height spectra of the charged particles escapedfrom a boron foil after the n (10B, α) 7Li reaction. The spectrum is always extendeddown to the lowest channels. As the charged particles are emitted randomly in space,their energy losses during the travelling through the material can be as high as theircomplete kinetic energy, that is why only some of them escape from the foil.
The absorption length for neutrons with a wavelength of 1.8 Å in a metallic10B-foil is 19.2 microns, whereas the ranges of the particles of the reactionn (10B,α) 7Li amount to 3.9 microns respectively 1.7 microns only. This means thatthe particle range is much shorter than the absorption length. Particles produceddeeper than this range cannot escape from the foil. The situation is similar for all
3.3-12
the other neutron converter foils. Therefore any combination of a particle detectorwith a foil leads to an arrangement with low detection efficiency. At a foil thicknesswhich is about 10% smaller than the range of the particle with the higher kineticenergy, the escape probability reaches a maximum as shown in Fig. (8). The foilthickness for a maximum neutron detection for commonly used converters isindicated in Tab (6).
Foil detectors are preferably used in neutronography and beam monitoring.For the latter a very thin layer of theabsorbing material is evaporated ontothe cathode inside a gas counter orionization chamber. For specialapplications, foil detectors areconstructed in combination withphotographic films, image plates,CCD’s and channel plates; in most casesGd foils are used because of their higherneutron efficiency and an easierhandling but the disadvantage is a highgamma sensitivity.
3.3-13
Figure 7: Calculated spectra of the charged particles escaped on thebackside from frontally irradiated 10B foils and produced in the reactionsn (10B, αα ) 7Li + 2.3 MeV + γγ(0.48 MeV) and n (10B, αα ) 7Li + 2.79 MeV. Theefficiency for 1.8 Å neutrons with the indicated foil thickness would be 1.0 %,respectively 5.5 %.
Figure 8: Escape probability of theparticles produced by a neutronreaction in a foil as function of itsthickness.
Other internal detectorsIn a special image plate for neutron detection the storage phosphor is mixed withGd2O3 11,13. By this means the efficiency can be increased: the conversionelectron does not have to escape from a foil but is stopped in the phosphor wherebythe latter will be excited. Such image plates are available in a size of 20 x 20 cm2.The dynamic of an image plate amounts to 1: 105. The spatial resolution of aneutron detector is of about 100 to 200 microns. For these reasons the device findsfavour in protein crystallography with its huge amount of interference spots. Butlike all the integrating detectors and particularly because of the incorporated Gdwith its higher atomic number, such a plate is rather gamma sensitive.
An excellent neutron detector which has not yet been constructed will beproposed here: The boron-diode. Boron is, like silicon or germanium, asemiconductor with a band gap of 1.4 eV. In the thirties, photographs used boronphotocells just as silicon cells are used today. With a surface barrier diode of boronor a boron diode doped with silicon, a high signal of each neutron could bemeasured. An energy of about 6 eV is necessary to produce one electron–hole pairin a semiconductor. 2.3 MeV are available in the n (10B, α) 7Li reaction , andtherefore a charge of 6 1014 [As] is generated. With an integrating operationamplifier and a capacitor of 1 pF in its loop an output signal of 60 mV could bemeasured. With a boron crystal thickness of only 60 microns, the neutron (1.8 Å )detection efficiency is of 97 %.
A position sensitive detector would be a CCD or a pixel device made of aboron crystal. and a position resolution of less than 10 microns would be achievedas the range of the α particle is only 3.9 microns. Moreover the gamma sensitivitywould be rather low as boron has a low atomic number.
FAQfind their answer in the textbook of Glenn F. Knoll 14.
3.3-14
Table 6: Foil thickness andmaximum detection efficiencyfor neutrons with a wavelengthof 1.8 Å for frontal irradiationand backside detection.
References[1] The stopping and range of ions in solids: J.F. Ziegler, J.P. Biersack andU. Littmark; Pergamon Press Inc., New York, USA (1985)
[2] H. Geiger, E. Rutherford and J.J. Thomson; Phil. Mag .XIII (1896) 392
[3] M.K. Kopp et al NIM 201 (1982) 395
[4] D. Feltin and B. Guérard, ILL NEWS 34 (2000) 8; Edit. G. Cicognani
http://www.ill.fr/News/34/034_NEWS.HTM
[5 ] G. Charpak ; NIM 62 (1968) 262
[6 ] A. Oed ; NIM A 263 (1988) 35
[7 ] F. Sauli ; NIM A 386 (1997) 531
[8 ] F. Sauli and A. Sharma ; Annual Rev. of Nucl. and Particle Sci. 49 ( 1999) 341
[9 ] A. Oed ; NIM A471 (2001) 109
[10 ] H. O. Anger ; Rev Sci. Instrum. 29 (1958) 27
[11 ] Ch. Rausch et al; SPIE Proc. 1737 (1992) 255
[12 ] Carel W.E. van Eijk ; NIM A 460 (2001) 1
[13] D.A.A. Myles et al; Physica B 241 - 243 ( 1998 ) 1122
[14] Radiation detection and measurement: Glenn F. Knoll; 3rd ed.1999;John Wiley& Sons, Inc, New York
3.3-15
Activation Table of the Elements
M. Johnson, S. A. Mason and R. B. Von Dreele
The best way to determine the activation of a sample is to measure it in situwith appropriate instruments. However in planning neutron experiments it may beuseful to know whether activation is likely to be significant. Properly estimatingactivation of a sample by a neutron beam requires knowledge of the neutron spec-trum, time of exposure, mass, isotopic composition, etc. The Table below allowsyou to calculate roughly the activation of a sample after it has been in a neutronbeam for one day and the amount of time for it to decay to 74Bq/g (i.e. 2nCi/g) orless, which is a typical limit for shipping a sample as “nonradioactive”. It also dis-plays the anticipated exposure you may receive when removing the sample from theinstrument. The entries in this table are derived from an approximate calculation(by M. Johnson) for decay times to 105 and to 104Bq/cm3 for 5cm3 pure solid sam-ples of the elements exposed to a neutron beam for 1day at an intensity comparableto that found on HIPD with LANSCE operating at 100µA. These calculations weremade for pulsed sources and may somewhat overestimate activation for reactors forsome elements (no epithermal neutrons). They are augmented by calculations fromNIST of the activation from a 1 day exposure to a 107n/s-cm2 reactor thermal beam(marked †). Storage time is the time required for a sample of the pure solid elementexposed to this “standard” neutron beam to decay to 74Bq/g or less. Prompt acti-vation gives the anticipated activation for the pure solid elements 2 min after theneutron exposure ceases. Contact dose is that expected from a 1g sample of thepure element from the prompt activation. Elements with a dash for the entries in allthree columns do not show any activation. Those marked with a single asterisk areradioactive before exposure to the neutron beam; apart from Tc and Pm, they are allα-particle emitters. Bismuth is a special case; it is stable before exposure to the
4.1-1
ACTIVATION AND SHIELDING
beam, but the activation product is an α-emitter. A typical sample activation calcu-lation may be found in the Web version of the table.
NB: The current Web version of this Table may be found on the ILL Web site: (http://www.ill.fr/YellowBook/D19/help/act_table.htm).
The original Table first appeared in the LANSCE Newsletter 12/1/92. We thankR. Pynn and V. T. Forsyth, for their cooperation. Comments to the present authors arevery welcome ([email protected], [email protected], [email protected]).
Ac actinium 227 * * *Al aluminium 26.982 21m 1900 2.0Am americium 243 * * *Sb antimony 121.75 520d 800 0.7Ar argon 39.948 19h 3500 3.0As arsenic 74.922 18d 8.4x104 7.3At astatine 210 * * *Ba barium 137.34 <150h <80 <0.1Bk berkelium 247 * * *Be beryllium 9.012 - - -Bi bismuth 208.980 ** ** **B boron 10.811 - - -Br bromine 79.909 18d 1.4x104 12†
Cd cadmium 112.40 190d 370 0.3Ca calcium 40.08 - - -Cf californium 249 * * *C carbon 12.011 - - -Ce cerium 140.12 <86h <40 <0.1Cs cesium 132.905 54h 4.6x105 400Cl chlorine 35.453 <2.8h <80 <0.1Cr chromium 51.996 <61d <40 <0.1Co cobalt 58.933 24y 5.2x104 45†
Cu copper 63.54 7.4d 1.0x104 8.5Cm curium 247 * * *Dy dysprosium 162.50 52h 5.0x105 430†
4.1-2
Symbol/Name Mass Storage Prompt activation Contact dose at 2.54cmTime 1 unit=37Bq/g 1 unit=10µGy/hr/g
(= 1nCi/g) (=1mr/hr/g, at 1 inch)
D deuterium 2.015 - - -Es einsteinium 254 * * *Er erbium 167.26 78d 600 0.5Eu europium 151.96 50y 2200 1.9†
Fm fermium 253 * * *F fluorine 18.998 - - -Fr francium 223 * * *Gd gadolinium 157.25 11d 7400 6.4Ga gallium 69.72 8d 3.2x104 27†
Ge germanium 72.59 <6d 1100 1.0†
Au gold 196.967 29d 3000 2.5Hf hafnium 178.49 1.6y 620 0.5He helium 4.003 - - -Ho holmium 164.930 20d 2.8x104 24†
H hydrogen 1.008 - - -In indium 114.82 12d 1.1x104 9.5†
I iodine 126.904 7h 1.2x105 100Ir iridium 192.2 4.2y 5.0x104 43†
Fe iron 55.847 - - -Kr krypton 83.80 42h 3200 2.8†
La lanthanum 138.91 22d 1.9x104 16Pb lead 207.19 - - -Li lithium 6.939 - - -Lu lutetium 174.97 1.8y 1.4x104 12†
Mg magnesium 24.312 - - -Mn manganese 54.938 38h 1.1x105 95Md mendelevium 256 * * *Hg mercury 200.59 24d 700 0.6Mo molybdenum 95.94 30d 430 0.4Nd neodymium 144.24 15h 1200 1.0Ne neon 20.183 - - -Np neptunium 237 * * *Ni nickel 58.71 <5.5h <30 <0.1Nb niobium 92.906 80m 2.0x104 17N nitrogen 14.007 - - -
4.1-3
Symbol/Name Mass Storage Prompt activation Contact dose at 2.54cmTime 1 unit=37Bq/g 1 unit=10µGy/hr/g
(= 1nCi/g) (=1mr/hr/g, at 1 inch)
Os osmium 190.2 41d 2300 2.0†
O oxygen 15.999 - - -Pd palladium 106.4 9d 7.1x104 60P phosphorus 30.974 - - -Pt platinum 195.09 20d 230 0.2Pu plutonium 242 * * *Po polonium 210 * * *K potassium 39.102 <38h <300 <0.3Pr praseodymium140.907 11d 2.0x104 17Pm promethium 147 * * *Pa proctactinium 231 * * *Ra radium 226 * * *Rn radon 222 * * *Re rhenium 186.2 53d 4.9x104 42Rh rhodium 102.905 2h 2.6x104 22†
Rb rubidium 85.47 56d 1800 1.6Ru ruthenium 101.07 106d 230 0.2Sm samarium 150.35 35d 6200 5.4Sc scandium 44.956 <1.8y <90 <0.1Se selenium 78.96 10h 4900 4.2†
Si silicon 28.086 - - -Ag silver 107.870 7.4y 1.6x104 14†
Na sodium 22.991 5.5d 5700 5.0Sr strontium 87.62 <25h <100 <0.1S sulphur 32.064 - - -Ta tantalum 180.948 3y 1600 1.4Tc technetium 98 * * *Te tellurium 127.60 96h 2600 2.2Tb terbium 158.924 2.1y 3300 2.8Tl thallium 204.37 41m 460 0.4Th thorium 232.038 * * *Tm thulium 168.934 3.7y 7700 6.7†
Sn tin 118.69 <50d <40 <0.1Ti titanium 47.90 - - -
4.1-4
Symbol/Name Mass Storage Prompt activation Contact dose at 2.54cmTime 1 unit=37Bq/g 1 unit=10µGy/hr/g
(= 1nCi/g) (=1mr/hr/g, at 1 inch)
W tungsten 183.85 15d 3.7x104 32U uranium 238.03 * * *V vanadium 50.942 48m 4.7x105 41Xe xenon 131.30 7d 3200 2.8Yb ytterbium 173.04 275d 780 0.7Y yttrium 88.905 24d 1000 0.9Zn zinc 65.37 5d 1600 1.4Zr zirconium 91.22 79h <40 <0.1
4.1-5
Symbol/Name Mass Storage Prompt activation Contact dose at 2.54cmTime 1 unit=37Bq/g 1 unit=10µGy/hr/g
(= 1nCi/g) (=1mr/hr/g, at 1 inch)
SHIELDING OF RADIATIONS
H.G. Börner and J. Tribolet
Shielding of photons The decrease in intensity of a parallel beam of photons traversing an absorber ofthickness d is given by
Φ = Φ0 e −Σd
where Φ0 and Φ are the beam intensity before and after passing through theabsorber, Σ is the attenuation coefficient for photons of energy E and for a givenmaterial. It should be noted that the attenuation calculated by the above formulagives only the decrease in intensity of the original beam. The total radiationdownstream of the absorber is larger, due to the presence of scattered photons, andto the creation of secondary photons by a variety of processes, includingfluorescence and positron annihilation radiation.
The graphs in Figure 1 give values for the attenuation coefficient Σ (expressedin cm-1) for some commonly used materials like iron, lead, aluminium andconcrete. One can deduce that one needs roughly 5 cm of lead or 50 cm of concrete,respectively, to attenuate a 10 MeV photon beam (a typical primary gamma rayenergy produced in neutron capture) by an order of magnitude.
Shielding of neutronsConcerning the attenuation of neutrons one has to distinguish between twocategories: Thermal and subthermal neutrons on one hand and epithermal and fastneutrons on the other one.
Thermal and subthermal neutrons can easily be stopped by capture inmaterials with high neutron capture cross sections like cadmium, gadolinium,boron and lithium. Generally a rather thin layer of absorbant material is sufficientto stop such neutron beams. However, in the case of cadmium and gadolinium theabsorption of neutrons is accompanied by strong emission of capture gamma rays.Therefore additional shielding is necessary to attenuate the gamma rays (seeabove). This phenomenon does not exist for lithium and is much less pronouncedfor boron. Consequently the latter materials are preferably used, especially on theoutside of heavy shielding in experimental areas.
4.2-1
4.2-2
Figure 1: Attenuation of photons.
To attenuate neutrons of higher energies is by far more complex because theycannot be stopped directly via capture and have first to be slowed down. A precisedetermination of a specific protection demands generally more complicatedcalculations using Monte Carlo methods. For neutrons in the MeV region, threetypes of materials are generally added together: dense materials (inelasticcollisions), hydrogenated materials (moderation), and absorbing materials(capture). An example is “heavy concrete” in which one finds iron, hydrogen andboron.
Specific examples:a) Neutron guides: In the parts which are sufficiently distant from the reactorneutrons are stopped by one to two layers (concerning ILL standards) of B4C.
b) Primary casemates: Typically 80 to 90 cm of heavy concrete are used (at ILL).This insures simultaneous protection against neutrons and gamma rays.
4.2-3
Figure 2: High energy part of the spectral distribution of neutrons emergingfrom a standard thermal beam tube.
c) Thermal neutron beam tubes: An example for the high energy part of the spectraldistribution of neutrons in a standard ‘thermal beam tube’ of ILL is shown inFigure 2. Optimized shielding against neutrons with such spectral distributiondepends on the space available:
• If 10 cm: Best attenuation is obtained by using borated polyethylene(attenuation by factor 25)
• If 20 cm: Best attenuation is obtained by using 2 cm of iron, followed by 18cm of borated polyethylene (attenuation by factor 310). However, the use of 20cm of borated polyethylene is only about 10 percent less efficient.
• If 30 cm: Best attenuation is obtained by using 12 cm of iron, followed by 18cm of borated polyethylene ( attenuation by factor 3400).
4.2-4
The International Commission on Radiation Units and Measurements (ICRU)recommends the use of SI units. Therefore, we list SI units first, followed by cgs (orother common) units in parentheses, where they differ.
5.1-1
MISCELLANEOUS
Radioactivity and RadiationProtection
5.1-2
5.2-1
Physical Constants
5.2-2
Physical Properties of the ElementsTable 5.3 lists several important properties of the elements.
Data were taken mostly from D. R. Lide, Ed., CRC Handbook of Chemistry andPhysics, 80th ed. (CRC Press, Boca Raton, Florida, 1999). Atomic weights apply toelements as they exist naturally on earth; values in parentheses are the mass num-bers for the longest-lived isotopes. Some uncertainty exists in the last digit of eachatomic weight. Specific heats are given for the elements at 25°C and a pressure of100 kPa. Densities for solids and liquids are given as specific gravities at 20°Cunless otherwise indicated by a superscript temperature (in°C); densities for thegaseous elements are given in g/cm3 for the liquids at their boiling points.
5.3-1
5.3-2
5.3-3
5.3-4
This table is adapted with permission from the 2001 web edition of the X-Ray DataBooklet (http://xdb.lbl.gov).
5.4
Disclaimer
Neither ILL nor any of its employees, nor any of the contributors to thepresent document makes any warranty, express or implied, or assumes any legalliability or responsibility for the accuracy, completeness, or usefulness of anyinformation, apparatus, product or process disclosed, or represents that its usewould not infringe privately owned rights.
The user accepts sole responsibility and any risks associated with the use andresults of material in this booklet, irrespective of the purpose to which such use orresults are applied.
